Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1580 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,27,30,42)(20,25,32,37)(22,35,28,26)(24,38,36,31)(29,43,33,46)(34,48,44,41)(39,47,40,45);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(18,23)(20,32)(24,36)(26,35)(27,42)(29,33)(34,44)(45,47);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,19,28,38,21,30,22,31)(18,24,35,42,23,36,26,27)(20,33,44,45,32,29,34,47)(25,40,48,43,37,39,41,46);(2,6)(7,8)(11,16)(13,15)(17,20)(18,25)(19,29)(21,32)(22,34)(23,37)(24,39)(26,41)(27,43)(28,44)(30,33)(31,45)(35,48)(36,40)(38,47)(42,46);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(20,32)(25,37)(29,33)(34,44)(39,40)(41,48)(43,46)(45,47)Polycyclic presentation:
2^9|a^2=i,c^2=fg,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6934524);