Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324436 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1789 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(17,18)(19,23)(20,31)(21,25)(22,27)(24,28)(26,30)(29,36)(32,44)(33,35)(34,39)(37,41)(38,42)(40,43)(45,46)(47,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,27,26)(20,28,33,42)(21,29,34,43)(24,35,38,31)(25,36,39,40)(32,41,45,48)(37,46,47,44);(2,6)(5,7)(9,11)(10,13)(12,16)(14,15)(17,20,30,42,22,33,19,28)(18,24,23,35,27,38,26,31)(21,32,43,48,34,45,29,41)(25,37,36,46,39,47,40,44);(3,8)(5,7)(10,14)(11,16)(17,21,22,34)(18,25,27,39)(19,29,30,43)(20,32,33,45)(23,36,26,40)(24,37,38,47)(28,41,42,48)(31,44,35,46);(2,6)(5,7)(10,14)(13,15)(21,34)(25,39)(29,43)(32,45)(36,40)(37,47)(41,48)(44,46)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgi,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7234003);