Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #45412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,27,42,26,31,25,32)(23,33,41,28,35,29,36,45)(24,30,43,52,39,49,40,48)(34,47,51,44,38,46,37,50);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,26,35)(22,28,31,45)(24,37,39,51)(25,41,27,36)(29,42,33,32)(30,47,49,46)(34,43,38,40)(44,52,50,48);(1,2,3,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,23)(22,29)(24,38)(25,36)(26,35)(27,41)(28,32)(30,47)(31,33)(34,39)(37,43)(40,51)(42,45)(44,52)(46,49)(48,50);(8,16)(11,20)(13,15)(17,19)(21,24,27,43,26,39,25,40)(22,30,42,52,31,49,32,48)(23,34,41,51,35,38,36,37)(28,44,29,46,45,50,33,47);(6,12)(7,14)(11,20)(13,15)(22,31)(23,35)(24,39)(32,42)(36,41)(40,43)(44,50)(46,47)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=gi,d^2=fi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6949413);