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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1733 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)(21,22,26,30)(23,31,34,27)(24,35,38,44)(25,39,42,29)(28,36,46,32)(33,43,40,47)(37,52,41,51)(45,49,48,50);(2,3)(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,14)(18,20)(21,23,26,34)(22,27,30,31)(24,36,38,32)(25,40,42,33)(28,44,46,35)(29,47,39,43)(37,49,41,50)(45,51,48,52);(1,2,4,3)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,14,20)(21,24,23,32,26,38,34,36)(22,28,27,35,30,46,31,44)(25,41,33,50,42,37,40,49)(29,48,43,52,39,45,47,51);(5,9,10,8)(6,12,13,18)(7,16,17,15)(11,19,14,20)(21,25,26,42)(22,29,30,39)(23,33,34,40)(24,37,38,41)(27,43,31,47)(28,45,46,48)(32,49,36,50)(35,51,44,52);(1,3,4,2)(7,17)(11,14)(15,16)(19,20)(25,42)(29,39)(33,40)(37,41)(43,47)(45,48)(49,50)(51,52)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgh,d^2=f,e^2=hi,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, 6722920);