Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500755 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2668 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2657 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,15)(4,21)(5,11)(6,13)(7,12)(8,14)(9,19)(10,23)(16,34)(17,43)(18,33)(20,32)(22,51)(24,41)(25,56)(26,39)(27,38)(28,40)(29,36)(30,45)(31,48)(35,53)(37,47)(42,46)(44,57)(49,58)(50,54)(52,60)(55,61)(59,63)(62,64);(1,3)(2,9)(4,16)(5,25)(6,20)(7,19)(8,18)(10,34)(11,37)(12,33)(13,15)(14,32)(17,26)(21,42)(22,52)(23,49)(24,45)(27,47)(28,48)(29,46)(30,36)(31,38)(35,60)(39,56)(40,43)(41,58)(44,53)(50,57)(51,59)(54,64)(55,63)(61,62);(1,4)(2,10)(3,16)(5,22)(6,23)(7,29)(8,24)(9,30)(11,35)(12,21)(13,41)(14,36)(15,42)(17,44)(18,45)(19,46)(20,49)(25,52)(26,53)(27,55)(28,54)(31,57)(32,34)(33,58)(37,62)(38,51)(39,50)(40,61)(43,60)(47,63)(48,64)(56,59);(1,5,8,28)(2,11,14,40)(3,17,18,47)(4,22,24,54)(6,27,7,26)(9,31,32,56)(10,35,36,61)(12,39,13,38)(15,43,33,37)(16,44,45,63)(19,48,20,25)(21,50,41,51)(23,55,29,53)(30,57,34,59)(42,60,58,62)(46,64,49,52);(3,18)(9,32)(15,33)(16,45)(17,47)(19,20)(25,48)(30,34)(31,56)(37,43)(42,58)(44,63)(46,49)(52,64)(57,59)(60,62)Polycyclic presentation:
2^9|d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=df,e^b=ei,f^b=fi,g^b=gi,g^c=gi,f^d=fi,g^d=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72306456, #72306460, #72306467, … |
| maximal normal subgroup of | #72306456, #72306460, #72306467, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7407952);