Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201968 |
| 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 | #60970 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
980 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)(3,13)(4,18)(5,10)(6,11)(7,12)(8,15)(9,20)(14,28)(16,24)(17,23)(19,25)(21,26)(22,31)(27,30)(29,32)(33,34,38,43,39,42,37,44)(35,45,48,55,49,58,47,40)(36,50,53,41,51,56,52,57)(46,54,62,63,60,64,61,59);(1,3,5,15,7,17,6,16)(2,8,11,13,12,24,10,23)(4,14,20,30,21,31,19,29)(9,22,18,27,25,28,26,32)(33,35)(34,40,42,55)(36,46)(37,47)(38,48)(39,49)(41,54,57,64)(43,45,44,58)(50,59,56,63)(51,60)(52,61)(53,62);(1,4)(2,9)(3,14)(5,20)(6,19)(7,21)(8,22)(10,26)(11,18)(12,25)(13,27)(15,30)(16,29)(17,31)(23,32)(24,28)(33,36)(34,41)(35,46)(37,53)(38,52)(39,51)(40,54)(42,57)(43,50)(44,56)(45,59)(47,62)(48,61)(49,60)(55,64)(58,63);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,21,20)(8,23,24,13)(9,18,25,26)(14,29,31,30)(22,27,28,32)(34,42)(36,51)(40,55)(43,44)(45,58)(46,60)(52,53)(61,62)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=hi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,f^b=fi,d^c=dh,e^c=ei,f^c=fh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 109318);