Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #28182 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2535 |
| 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 | |
1592 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)(7,19)(8,25,30,44)(9,31)(10,17)(11,16)(12,18)(13,22)(14,29,41,28)(15,32)(20,35,47,53)(21,48)(23,38)(24,37)(26,55,52,39)(27,57,60,40)(33,43)(34,42)(36,51)(45,49)(46,63,65,68)(50,62)(54,59,64,58)(56,61,67,66);(1,2,4,3)(5,7,10,22,12,24,11,23)(6,13,16,19,18,38,17,37)(8,26,29,20,30,52,28,47)(9,21,32,49,34,51,33,50)(14,39,44,35,41,55,25,53)(15,36,31,45,43,48,42,62)(27,56,59,46,60,67,58,65)(40,63,64,61,57,68,54,66);(5,8)(6,14)(7,20)(9,27)(10,28)(11,29)(12,30)(13,35)(15,40)(16,25)(17,44)(18,41)(19,39)(21,46)(22,26)(23,52)(24,47)(31,54)(32,58)(33,59)(34,60)(36,61)(37,55)(38,53)(42,64)(43,57)(45,63)(48,66)(49,56)(50,67)(51,65)(62,68);(1,3,4,2)(5,9,11,33,12,34,10,32)(6,15,17,42,18,43,16,31)(7,21,23,50,24,51,22,49)(8,27,29,59,30,60,28,58)(13,36,37,62,38,48,19,45)(14,40,44,64,41,57,25,54)(20,46,52,67,47,65,26,56)(35,61,55,68,53,66,39,63)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,f^c=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272238);