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 | #42908 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1784 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,23,30,40)(20,25,32,43)(21,34,36,27)(24,35,42,33)(28,44,29,38)(31,48,47,39)(37,46,41,45);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,29)(21,31)(22,30)(24,41)(25,38)(26,40)(27,39)(28,32)(33,46)(34,48)(35,45)(36,47)(37,42)(43,44);(9,11)(10,13)(12,16)(14,15)(17,20)(18,24)(19,28)(21,35)(22,32)(23,37)(25,34)(26,42)(27,43)(29,30)(31,46)(33,36)(38,39)(40,41)(44,48)(45,47);(3,8)(5,7)(11,16)(13,15)(17,20)(18,25)(19,29)(21,33)(22,32)(23,38)(24,34)(26,43)(27,42)(28,30)(31,46)(35,36)(37,39)(40,44)(41,48)(45,47);(10,14)(13,15)(18,26)(19,30)(20,32)(27,34)(31,47)(33,35)(37,41)(38,44)Polycyclic presentation:
2^9|a^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^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, 6901991);