Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396726 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14737 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13)(3,15,19,33)(4,10,25,38)(5,26,28,40)(6,14,8,12)(9,20,32,18)(11,29,39,27)(16,41,21,50)(17,46,48,31)(22,52,54,60)(23,37,24,36)(30,45,34,44)(35,55,59,53)(42,49,57,47)(43,58,51,56)(61,63,64,62);(1,3,8,20,7,19,6,18)(2,9,14,33,13,32,12,15)(4,21,23,45,25,16,24,44)(5,17,29,49,28,48,27,47)(10,34,36,50,38,30,37,41)(11,31,40,57,39,46,26,42)(22,51,53,63,54,43,55,62)(35,58,52,64,59,56,60,61);(1,4,6,23,7,25,8,24)(2,10,12,36,13,38,14,37)(3,16,18,44,19,21,20,45)(5,22,27,53,28,54,29,55)(9,30,15,41,32,34,33,50)(11,35,26,52,39,59,40,60)(17,43,47,62,48,51,49,63)(31,56,42,61,46,58,57,64);(1,5)(2,11)(3,17)(4,22)(6,27)(7,28)(8,29)(9,31)(10,35)(12,26)(13,39)(14,40)(15,42)(16,43)(18,47)(19,48)(20,49)(21,51)(23,53)(24,55)(25,54)(30,56)(32,46)(33,57)(34,58)(36,52)(37,60)(38,59)(41,61)(44,62)(45,63)(50,64)Polycyclic presentation:
2^9|a^2=i,b^2=ef,c^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,f^c=fi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303923);