Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453995 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2812 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16249 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2138 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)(4,20)(5,11)(6,13)(7,12)(8,14)(9,18)(10,23)(16,30)(17,47)(19,40)(21,53)(22,36)(24,35)(25,56)(26,41)(27,39)(28,32)(29,38)(31,49)(33,46)(34,55)(37,54)(42,45)(43,51)(44,58)(48,64)(50,61)(52,59)(57,60)(62,63);(1,3)(2,9)(4,16)(5,17)(6,28)(7,18)(8,19)(10,33)(11,31)(12,40)(13,15)(14,32)(20,42)(21,44)(22,52)(23,45)(24,46)(25,57)(26,50)(27,51)(29,49)(30,36)(34,62)(35,59)(37,60)(38,43)(39,61)(41,47)(48,55)(53,64)(54,63)(56,58);(1,4)(2,10)(3,16)(5,25)(6,23)(7,22)(8,24)(9,30)(11,37)(12,20)(13,35)(14,36)(15,42)(17,48)(18,52)(19,46)(21,26)(27,55)(28,45)(29,54)(31,60)(32,33)(34,38)(39,53)(40,59)(41,56)(43,62)(44,49)(47,58)(50,63)(51,57)(61,64);(1,5)(2,11)(3,17)(4,21)(6,29)(7,26)(8,27)(9,31)(10,34)(12,41)(13,38)(14,39)(15,43)(16,44)(18,50)(19,51)(20,37)(22,55)(23,25)(24,54)(28,49)(30,58)(32,61)(33,62)(35,53)(36,56)(40,47)(42,60)(45,57)(46,63)(48,52)(59,64)Polycyclic presentation:
2^9|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,e^b=ei,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361191);