Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418473 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2673 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15320 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,13,8,29)(3,14,17,33,18,34,28,9)(4,19,21,49,22,10,23,37)(5,24,26,11,27,39,30,40)(15,41,43,35,44,59,54,31)(16,45,25,57,47,32,46,38)(20,50,52,48,53,56,55,36)(42,60,51,61,64,58,63,62);(1,3)(2,9)(4,15)(5,25)(6,28)(7,18)(8,17)(10,35)(11,38)(12,34)(13,33)(14,29)(16,26)(19,31)(20,51)(21,54)(22,44)(23,43)(24,45)(27,46)(30,47)(32,39)(36,61)(37,41)(40,57)(42,52)(48,62)(49,59)(50,60)(53,63)(55,64)(56,58);(1,4,8,23,7,22,6,21)(2,10,29,49,13,19,12,37)(3,15,17,43,18,44,28,54)(5,20,30,55,27,53,26,52)(9,31,14,41,33,35,34,59)(11,36,24,56,40,48,39,50)(16,42,25,51,47,64,46,63)(32,58,38,62,45,60,57,61);(1,5,6,26,7,27,8,30)(2,11,12,39,13,40,29,24)(3,16,17,25,18,47,28,46)(4,20,21,52,22,53,23,55)(9,32,14,38,33,45,34,57)(10,36,37,50,19,48,49,56)(15,42,43,51,44,64,54,63)(31,58,41,62,35,60,59,61)Polycyclic presentation:
2^9|a^2=e,c^2=ef,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 331512);