Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #397514 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
996 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2,4,3)(5,6)(7,19)(8,24)(9,30)(10,18)(11,16)(12,17)(13,22)(14,27)(15,32)(20,46)(21,49)(23,43)(25,35)(26,38)(28,39)(29,40)(31,42)(33,41)(34,37)(36,51)(44,48)(45,50)(47,66)(52,55)(53,58)(54,64)(56,59)(57,62)(60,63)(61,68)(65,67);(1,3,4,2)(5,7,12,23)(6,13,17,37)(8,25,28,20)(9,21,33,51)(10,34,11,22)(14,35,40,46)(15,36,42,49)(16,43,18,19)(24,52,39,44)(26,54,57,47)(27,48,29,55)(30,45,41,63)(31,60,32,50)(38,61,62,65)(53,66,59,64)(56,67,58,68);(2,3)(5,8,10,27,12,28,11,29)(6,14,16,24,17,40,18,39)(7,20,22,48,23,25,34,55)(9,26,31,56,33,57,32,58)(13,35,19,44,37,46,43,52)(15,38,41,53,42,62,30,59)(21,47,50,67,51,54,60,68)(36,61,45,64,49,65,63,66);(5,9,10,31,12,33,11,32)(6,15,16,41,17,42,18,30)(7,21,34,60,23,51,22,50)(8,26,27,56,28,57,29,58)(13,36,43,63,37,49,19,45)(14,38,24,53,40,62,39,59)(20,47,55,68,25,54,48,67)(35,61,52,66,46,65,44,64)Polycyclic presentation:
2^9|a^2=efi,b^2=ef,c^2=e,d^2=hi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 320327);