Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16797 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1069 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)(3,14)(4,10)(5,24)(6,12)(7,29)(8,13)(9,17)(11,27)(15,41)(16,45)(18,33)(19,51)(20,53)(21,36)(22,39)(23,37)(25,40)(26,38)(28,32)(30,44)(31,50)(34,49)(35,56)(42,48)(43,63)(46,61)(47,57)(52,59)(54,58)(55,62)(60,64);(1,3)(2,9,13,14)(4,19)(5,16)(6,28)(7,18)(8,17)(10,34,37,51)(11,31,38,42)(12,33,29,32)(15,21)(20,52)(22,44)(23,49)(24,57,40,45)(25,47)(26,50)(27,48)(30,39,41,36)(35,61,62,63)(43,56)(46,55)(53,60,58,59)(54,64);(1,4,8,23)(2,10,13,37)(3,15,17,44)(5,20,25,54)(6,22,7,21)(9,30,14,41)(11,35,38,62)(12,39,29,36)(16,46,47,43)(18,49,28,19)(24,58,40,53)(26,56,27,55)(31,60,42,59)(32,51,33,34)(45,63,57,61)(48,64,50,52);(1,5)(2,11)(3,16)(4,20)(6,27)(7,26)(8,25)(9,31)(10,35)(12,40)(13,38)(14,42)(15,43)(17,47)(18,50)(19,52)(21,56)(22,55)(23,54)(24,29)(28,48)(30,59)(32,57)(33,45)(34,61)(36,58)(37,62)(39,53)(41,60)(44,46)(49,64)(51,63)Polycyclic presentation:
2^9|b^2=egh,c^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,g^c=gi,h^c=hi,f^d=fi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72287064, #72287280, #72287312, … |
| maximal normal subgroup of | #72287064, #72287280, #72287312, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378635);