Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #249839 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#41 | C2×D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
890 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 890 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 96 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 96 points:
(1,2,7,12)(3,8,15,25)(4,16,18,28)(5,10,20,24)(6,21,14,11)(9,22,27,19)(13,26,29,31)(17,32,30,23)(33,34)(35,46)(36,52)(37,58)(38,62)(39,44)(40,45)(41,49)(42,56)(43,59)(47,75)(48,64)(50,66)(51,65)(53,73)(54,84)(55,70)(57,69)(60,72)(61,71)(63,77)(67,81)(68,87)(74,78)(76,93)(79,90)(80,89)(82,85)(83,88)(86,92)(91,96)(94,95);(1,3,7,15)(2,8,12,25)(4,13,18,29)(5,6,20,14)(9,23,27,32)(10,21,24,11)(16,31,28,26)(17,22,30,19)(33,35)(34,41)(36,53)(37,48)(38,50)(39,49)(40,51)(42,67)(43,64)(44,65)(45,66)(46,62)(47,55)(52,63)(54,83)(56,77)(57,81)(58,74)(59,80)(60,78)(61,79)(68,91)(69,75)(70,73)(71,89)(72,90)(76,86)(82,88)(84,93)(85,96)(87,95)(92,94);(1,4,7,18)(2,9,12,27)(3,13,15,29)(5,17,20,30)(6,22,14,19)(8,23,25,32)(10,26,24,31)(11,28,21,16)(33,36)(34,42)(35,47)(37,54)(38,55)(39,56)(40,57)(41,63)(43,68)(44,69)(45,70)(46,73)(48,76)(49,53)(50,81)(51,77)(52,62)(58,82)(59,85)(60,86)(61,87)(64,88)(65,67)(66,75)(71,92)(72,84)(74,93)(78,95)(79,96)(80,83)(89,91)(90,94);(1,5,7,20)(2,10,12,24)(3,14,15,6)(4,17,18,30)(8,11,25,21)(9,26,27,31)(13,19,29,22)(16,23,28,32)(33,37,40,61)(34,43,45,72)(35,48,51,79)(36,54,57,87)(38,60,39,59)(41,64,66,90)(42,68,70,84)(44,71,62,58)(46,74,65,89)(47,76,77,96)(49,80,50,78)(52,82,69,92)(53,83,81,95)(55,86,56,85)(63,88,75,94)(67,91,73,93)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=hi,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,c^b=cg,d^b=di,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fhi,g^c=gh,e^d=eh,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 157073);