Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682596 |
| 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 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1840 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)(11,13)(12,14)(15,16)(17,18,23,27)(19,24,30,40)(20,31,34,43)(21,26,22,37)(25,36,44,33)(28,45,32,48)(29,39,35,41)(38,46,42,47);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,32)(21,35)(22,29)(23,30)(25,42)(26,41)(27,40)(28,34)(31,48)(33,46)(36,47)(37,39)(38,44)(43,45);(10,14)(11,15)(17,20,21,36,23,34,22,33)(18,25,37,43,27,44,26,31)(19,28,35,46,30,32,29,47)(24,38,39,48,40,42,41,45);(2,6)(5,7)(17,21,23,22)(18,26,27,37)(19,29,30,35)(20,33,34,36)(24,39,40,41)(25,43,44,31)(28,46,32,47)(38,45,42,48);(3,8)(5,7)(11,15)(13,16)(19,30)(20,34)(24,40)(25,44)(29,35)(31,43)(33,36)(39,41)Polycyclic presentation:
2^9|a^2=i,c^2=gh,d^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=efh,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6669330);