Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 | #37701 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6)(7,14)(8,12)(9,18)(10,13)(11,17)(15,20)(16,19)(21,22,27,31)(23,28,26,41)(24,34,37,51)(25,30,40,47)(29,32,44,36)(33,42,38,45)(35,50,39,52)(43,48,46,49);(2,3)(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,14)(18,20)(21,23,27,26)(22,28,31,41)(24,32,37,36)(25,38,40,33)(29,34,44,51)(30,45,47,42)(35,49,39,48)(43,52,46,50);(1,2,4,3)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,14,20)(21,24,26,36,27,37,23,32)(22,29,28,34,31,44,41,51)(25,39,38,48,40,35,33,49)(30,46,42,52,47,43,45,50);(5,9,10,8)(6,12,13,18)(7,16,17,15)(11,19,14,20)(21,25)(22,30)(23,33)(24,35)(26,38)(27,40)(28,42)(29,43)(31,47)(32,48)(34,50)(36,49)(37,39)(41,45)(44,46)(51,52);(1,3,4,2)(7,17)(11,14)(15,16)(19,20)(25,40)(30,47)(33,38)(35,39)(42,45)(43,46)(48,49)(50,52)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgh,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6722919);