Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | #45385 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1495 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)(17,18,23,25,21,27,22,28)(19,29,36,40,31,24,32,38)(20,26,37,41,34,43,35,44)(30,42,46,48,33,39,45,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,33)(21,31)(22,32)(23,36)(25,38)(26,42)(27,29)(28,40)(30,34)(35,46)(37,45)(39,43)(41,48)(44,47);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(18,25)(22,23)(24,38)(26,41)(27,28)(29,40)(32,36)(35,37)(39,47)(42,48)(43,44)(45,46);(1,4,5,3)(2,7,6,8)(11,16)(13,15)(17,20)(18,26)(19,30)(21,34)(22,35)(23,37)(24,39)(25,41)(27,43)(28,44)(29,42)(31,33)(32,45)(36,46)(38,47)(40,48);(10,14)(13,15)(19,31)(20,34)(24,29)(26,43)(32,36)(35,37)(38,40)(41,44)Polycyclic presentation:
2^9|a^2=gi,c^2=fi,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,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, 6915593);