Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201978 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #3192 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
971 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,11,7,12,5,10)(3,13,15,23,17,26,16,8)(4,18,19,9,21,24,20,25)(14,22,29,27,30,32,28,31)(33,34)(35,44)(36,49,52,58)(37,53)(38,42)(39,43)(40,47)(41,51,57,50)(45,60,63,54)(46,56)(48,55)(59,62,64,61);(1,3)(2,8,12,23)(4,14)(5,16)(6,15)(7,17)(9,22,25,32)(10,26,11,13)(18,31,24,27)(19,29)(20,28)(21,30)(33,35,37,47,39,48,38,46)(34,40,42,44,43,56,53,55)(36,45,51,61,52,63,50,62)(41,54,58,59,57,60,49,64);(1,4)(2,9)(3,14)(5,20)(6,19)(7,21)(8,22)(10,18)(11,24)(12,25)(13,27)(15,29)(16,28)(17,30)(23,32)(26,31)(33,36)(34,41)(35,45)(37,51)(38,50)(39,52)(40,54)(42,58)(43,57)(44,59)(46,62)(47,61)(48,63)(49,53)(55,64)(56,60);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,21,20)(8,13,23,26)(9,24,25,18)(14,28,30,29)(22,31,32,27)(33,37,39,38)(34,42,43,53)(35,46,48,47)(36,50,52,51)(40,55,56,44)(41,49,57,58)(45,61,63,62)(54,59,60,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,d^b=dh,f^b=fi,d^c=dh,e^c=ei,f^c=fh,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 109372);