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 | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1584 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)(3,4)(9,10)(11,14)(12,13)(15,16)(17,18,22,27)(19,23,30,39)(20,25,32,43)(21,34,36,26)(24,35,42,33)(28,44,29,38)(31,48,47,40)(37,46,41,45);(1,2)(3,7)(4,8)(5,6)(10,13)(11,15)(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);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,20)(18,24)(19,28)(21,35)(22,32)(23,37)(25,34)(26,43)(27,42)(29,30)(31,46)(33,36)(38,40)(39,41)(44,48)(45,47);(1,4,5,3)(2,8,6,7)(10,13)(14,16)(17,20)(18,25)(19,29)(21,33)(22,32)(23,38)(24,34)(26,42)(27,43)(28,30)(31,46)(35,36)(37,40)(39,44)(41,48)(45,47);(2,6)(7,8)(19,30)(20,32)(23,39)(24,42)(25,43)(31,47)(33,35)(40,48)Polycyclic presentation:
2^9|a^2=i,c^2=f,d^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,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, 6679922);