Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,16)(11,14,12,15)(17,18)(19,29)(20,34)(21,27)(22,28)(23,38)(24,32)(25,35)(26,36)(30,40)(31,45)(33,41)(37,42)(39,48)(43,46)(44,47);(1,3)(2,5)(4,8)(6,7)(11,12)(14,15)(17,19,22,32)(18,24,28,29)(20,30,36,47)(21,31,23,33)(25,39,42,43)(26,40,34,44)(27,41,38,45)(35,46,37,48);(9,11,13,12)(10,14,16,15)(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,39)(26,27)(28,42)(29,43)(31,46)(32,47)(33,48)(34,38)(40,41)(44,45);(2,6)(5,7)(17,20)(18,26)(19,30)(21,37)(22,36)(23,35)(24,40)(25,27)(28,34)(29,44)(31,48)(32,47)(33,46)(38,42)(39,41)(43,45);(3,8)(5,7)(9,12,13,11)(10,15,16,14)(17,21,22,23)(18,27,28,38)(19,31,32,33)(20,35,36,37)(24,41,29,45)(25,26,42,34)(30,46,47,48)(39,40,43,44)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6754170);