Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499750 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23848 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2930 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1259 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)(3,13)(4,18)(5,22)(6,28)(7,11)(8,15)(9,19)(10,27)(12,26)(14,32)(16,50)(17,43)(20,52)(21,54)(23,39)(24,41)(25,40)(29,36)(30,38)(31,48)(33,55)(34,53)(35,56)(37,47)(42,44)(45,57)(46,60)(49,58)(51,59)(61,63)(62,64);(1,3,7,17)(2,8,11,31)(4,14,20,45)(5,23,24,55)(6,29,26,16)(9,32,34,57)(10,36,37,50)(12,39,28,33)(13,41,43,22)(15,47,48,27)(18,42,52,61)(19,49,53,64)(21,51,30,46)(25,44,56,63)(35,60,40,59)(38,58,54,62);(5,24)(8,31)(10,37)(15,48)(18,52)(19,53)(22,41)(23,55)(25,56)(27,47)(32,57)(33,39)(38,54)(44,63)(49,64)(59,60);(1,4)(2,9)(3,14)(5,25)(6,30)(7,20)(8,32)(10,38)(11,34)(12,40)(13,42)(15,49)(16,51)(17,45)(18,22)(19,27)(21,26)(23,44)(24,56)(28,35)(29,46)(31,57)(33,60)(36,58)(37,54)(39,59)(41,52)(43,61)(47,53)(48,64)(50,62)(55,63)Polycyclic presentation:
2^9|b^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,h^a=hi,c^b=cg,e^b=eh,h^b=hi,e^c=ei,e^d=eh,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72307706, #72307725, #72307931, … |
| maximal normal subgroup of | #72307706, #72307725, #72307931, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406947);