Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #222240 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| 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 | |
1397 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,3)(5,6)(7,18,22,37)(8,23)(9,29,32,43)(10,16)(11,34)(12,17)(13,21,38,33)(14,27)(15,30,41,31)(19,46,24,35)(20,36)(25,39,57,63)(26,40)(28,42)(44,51,59,48)(45,52)(47,66)(49,60)(50,61)(53,55,62,56)(54,64)(58,67)(65,68);(1,2,4,3)(5,7,10,33,12,22,11,21)(6,13,16,37,17,38,34,18)(8,24,26,48,28,19,27,51)(9,20,31,49,32,50,30,52)(14,35,23,44,40,46,42,59)(15,36,43,60,41,61,29,45)(25,54,56,68,57,47,55,67)(39,58,53,64,63,65,62,66);(5,8,12,28)(6,14,17,40)(7,19,22,24)(9,25,32,57)(10,27,11,26)(13,35,38,46)(15,39,41,63)(16,42,34,23)(18,44,37,59)(20,47,50,54)(21,51,33,48)(29,53,43,62)(30,56,31,55)(36,58,61,65)(45,64,60,66)(49,68,52,67);(1,3,4,2)(5,9)(6,15)(7,20)(8,25)(10,31)(11,30)(12,32)(13,36)(14,39)(16,43)(17,41)(18,45)(19,47)(21,52)(22,50)(23,53)(24,54)(26,56)(27,55)(28,57)(29,34)(33,49)(35,58)(37,60)(38,61)(40,63)(42,62)(44,64)(46,65)(48,68)(51,67)(59,66)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=i,d^2=efg,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,e^d=ei,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 279935);