Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394376 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14716 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2515 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)(4,10)(5,25)(6,13)(7,12)(8,14)(9,18)(11,26)(16,42)(17,45)(19,33)(20,32)(21,30)(22,51)(23,37)(24,38)(27,39)(28,36)(29,40)(31,47)(34,44)(35,53)(41,46)(43,55)(48,56)(49,57)(50,58)(52,60)(54,59)(61,63)(62,64);(1,3,6,18,8,19,7,20)(2,9,12,15,14,32,13,33)(4,21,23,44,24,48,28,16)(5,17,26,46,27,47,29,49)(10,34,36,30,38,42,37,56)(11,31,25,41,40,45,39,57)(22,50,52,62,54,63,53,43)(35,58,59,55,60,61,51,64);(1,4)(2,10)(3,16)(5,22)(6,28)(7,23)(8,24)(9,30)(11,35)(12,37)(13,36)(14,38)(15,34)(17,43)(18,48)(19,44)(20,21)(25,51)(26,53)(27,54)(29,52)(31,55)(32,56)(33,42)(39,59)(40,60)(41,58)(45,64)(46,63)(47,62)(49,50)(57,61);(1,5,8,27)(2,11,14,40)(3,17,19,47)(4,22,24,54)(6,26,7,29)(9,31,32,45)(10,35,38,60)(12,25,13,39)(15,41,33,57)(16,43,44,62)(18,46,20,49)(21,50,48,63)(23,52,28,53)(30,55,56,64)(34,58,42,61)(36,59,37,51)Polycyclic presentation:
2^9|b^2=e,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,e^c=ei,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72287064, #72287261, #72287280, … |
| maximal normal subgroup of | #72287064, #72287261, #72287280, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301573);