Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394088 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14715 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3852 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)(5,11)(6,27)(7,12)(8,13)(9,17)(15,40)(16,41)(18,31)(19,39)(20,29)(21,49)(22,35)(23,36)(24,52)(25,38)(26,37)(28,34)(30,44)(32,43)(33,51)(42,57)(45,60)(46,56)(47,55)(48,64)(50,58)(53,59)(54,63)(61,62);(1,3,8,19)(2,9,13,31)(4,20,23,47)(5,16,25,45)(6,18,7,17)(10,32,36,40)(11,30,38,56)(12,39,27,14)(15,28,43,22)(21,48,51,63)(24,46,26,44)(29,34,55,35)(33,57,49,61)(37,60,52,41)(42,53,62,50)(54,58,64,59);(1,4)(2,10)(3,15)(5,21)(6,22)(7,28)(8,23)(9,29)(11,33)(12,34)(13,36)(14,40)(16,42)(17,20)(18,47)(19,43)(24,50)(25,51)(26,53)(27,35)(30,54)(31,55)(32,39)(37,58)(38,49)(41,61)(44,48)(45,62)(46,63)(52,59)(56,64)(57,60);(1,5)(2,11)(3,16)(4,21)(6,24)(7,26)(8,25)(9,30)(10,33)(12,37)(13,38)(14,41)(15,42)(17,44)(18,46)(19,45)(20,48)(22,50)(23,51)(27,52)(28,53)(29,54)(31,56)(32,57)(34,58)(35,59)(36,49)(39,60)(40,61)(43,62)(47,63)(55,64)Polycyclic presentation:
2^9|b^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi,g^c=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301285);