Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470922 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
1653 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27)(19,28,32,38)(20,33,35,41)(21,25,29,26)(23,36,39,31)(24,40,42,34)(30,44,46,37)(43,47,48,45);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,29,36,22,32,21,31)(18,23,25,38,27,39,26,28)(20,30,40,47,35,46,34,45)(24,37,41,48,42,44,33,43);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,20,21,34,22,35,29,40)(18,24,26,33,27,42,25,41)(19,30,31,45,32,46,36,47)(23,37,28,43,39,44,38,48);(11,15)(13,16)(17,21,22,29)(18,25,27,26)(19,31,32,36)(20,34,35,40)(23,38,39,28)(24,41,42,33)(30,45,46,47)(37,48,44,43);(10,14)(13,16)(20,35)(24,42)(30,46)(33,41)(34,40)(37,44)(43,48)(45,47)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=f,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6380125);