Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| 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 | #38032 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,28)(20,33)(21,27)(22,24)(23,37)(25,32)(26,34)(29,44)(30,38)(31,42)(35,41)(36,40)(39,47)(43,45)(46,48);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(18,24)(20,34)(25,38)(27,37)(28,42)(29,45)(35,36)(46,47);(11,16)(13,15)(17,19,22,31)(18,25,24,38)(20,29,34,45)(21,32,23,30)(26,39,33,48)(27,42,37,28)(35,47,36,46)(40,43,41,44);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20)(18,26)(19,29)(21,36)(22,34)(23,35)(24,33)(25,39)(27,41)(28,43)(30,47)(31,45)(32,46)(37,40)(38,48)(42,44);(2,5)(6,8)(17,21,22,23)(18,27,24,37)(19,30,31,32)(20,35,34,36)(25,28,38,42)(26,40,33,41)(29,46,45,47)(39,44,48,43)Polycyclic presentation:
2^9|c^2=i,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,e^c=ei,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6678247);