Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32950 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,4)(5,6,11,17,12,16,10,30)(7,13,21,36,20,37,31,35)(8,22,25,39,26,50,24,14)(9,27,29,40,28,15,32,41)(18,42,45,33,44,59,54,60)(19,34,48,61,47,56,46,62)(23,51,53,57,52,49,55,38)(43,63,66,68,65,67,64,58);(1,2,3,4)(5,7,10,21,12,20,11,31)(6,13,17,35,16,37,30,36)(8,18,24,45,26,44,25,54)(9,19,29,46,28,47,32,48)(14,33,22,42,39,60,50,59)(15,34,40,61,27,56,41,62)(23,43,53,64,52,65,55,66)(38,58,49,63,57,68,51,67);(5,8,11,25,12,26,10,24)(6,14,30,50,16,39,17,22)(7,18,31,54,20,44,21,45)(9,23,32,55,28,52,29,53)(13,33,36,59,37,60,35,42)(15,38,41,51,27,57,40,49)(19,43,48,66,47,65,46,64)(34,58,62,67,56,68,61,63);(5,9)(6,15)(7,19)(8,23)(10,29)(11,32)(12,28)(13,34)(14,38)(16,27)(17,40)(18,43)(20,47)(21,46)(22,49)(24,53)(25,55)(26,52)(30,41)(31,48)(33,58)(35,61)(36,62)(37,56)(39,57)(42,63)(44,65)(45,64)(50,51)(54,66)(59,67)(60,68)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,c^2=fi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gi,e^b=ei,g^b=gi,h^b=hi,e^d=ei,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300232);