Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418530 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2676 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15263 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1641 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,15)(4,21)(5,25)(6,13)(7,12)(8,14)(9,18)(10,23)(11,28)(16,35)(17,38)(19,34)(20,33)(22,49)(24,40)(26,32)(27,39)(29,37)(30,41)(31,45)(36,51)(42,44)(43,56)(46,54)(47,58)(48,57)(50,60)(52,55)(53,61)(59,64)(62,63);(1,3)(2,9)(4,16)(5,26)(6,19)(7,18)(8,20)(10,35)(11,38)(12,33)(13,15)(14,34)(17,27)(21,42)(22,50)(23,47)(24,45)(25,54)(28,48)(29,44)(30,46)(31,37)(32,39)(36,60)(40,58)(41,57)(43,51)(49,59)(52,64)(53,63)(55,62)(56,61);(1,4)(2,10)(3,16)(5,22)(6,23)(7,29)(8,24)(9,31)(11,36)(12,21)(13,40)(14,37)(15,42)(17,43)(18,44)(19,47)(20,45)(25,55)(26,50)(27,51)(28,53)(30,52)(32,56)(33,58)(34,35)(38,62)(39,49)(41,61)(46,64)(48,63)(54,60)(57,59);(1,5,6,27,8,28,7,30)(2,11,12,39,14,25,13,41)(3,17,18,26,20,46,19,48)(4,22,23,51,24,53,29,52)(9,32,15,38,34,57,33,54)(10,36,21,49,37,55,40,61)(16,43,44,50,45,64,47,63)(31,56,42,62,35,59,58,60)Polycyclic presentation:
2^9|d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325726);