Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219491 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| 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 | |
1486 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)(8,25,30,41)(9,31)(10,17)(11,16)(12,18)(13,22)(14,28,42,29)(15,32)(20,46,49,35)(21,50)(23,38)(24,37)(26,39,48,60)(27,40,58,64)(33,44)(34,43)(36,53)(45,51)(47,62,67,68)(52,61)(54,57,63,56)(55,59,66,65);(1,2,4,3)(5,7,11,23,12,24,10,22)(6,13,17,37,18,38,16,19)(8,26,29,49,30,48,28,20)(9,21,33,52,34,53,32,51)(14,39,25,46,42,60,41,35)(15,36,43,61,44,50,31,45)(27,55,57,67,58,66,56,47)(40,62,54,65,64,68,63,59);(5,8)(6,14)(7,20)(9,27)(10,29)(11,28)(12,30)(13,35)(15,40)(16,25)(17,41)(18,42)(19,39)(21,47)(22,26)(23,48)(24,49)(31,54)(32,57)(33,56)(34,58)(36,59)(37,60)(38,46)(43,63)(44,64)(45,62)(50,65)(51,55)(52,66)(53,67)(61,68);(1,3,4,2)(5,9,10,32,12,34,11,33)(6,15,16,31,18,44,17,43)(7,21,22,51,24,53,23,52)(8,27,29,57,30,58,28,56)(13,36,19,45,38,50,37,61)(14,40,25,54,42,64,41,63)(20,47,26,55,49,67,48,66)(35,59,39,62,46,65,60,68)Polycyclic presentation:
2^9|a^2=h,b^2=ei,d^2=eh,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,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283390);