Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32947 |
| 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 | |
1766 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)(7,19)(8,25)(9,30,32,15)(10,17)(11,16)(12,18)(13,22)(14,28)(20,34)(21,44,50,62)(23,37)(24,36)(26,54,56,53)(27,40)(29,39)(31,41,33,42)(35,49,58,51)(38,55,59,57)(43,47)(45,60,65,68)(46,52)(48,61)(63,66,67,64);(1,2,3,4)(5,7,10,22,12,24,11,23)(6,13,16,19,18,37,17,36)(8,20,27,46,29,48,28,47)(9,21,33,51,32,50,31,49)(14,34,39,43,40,61,25,52)(15,35,42,44,30,58,41,62)(26,45,57,66,56,65,55,64)(38,60,54,63,59,68,53,67);(5,8,10,27,12,29,11,28)(6,14,16,39,18,40,17,25)(7,20,22,46,24,48,23,47)(9,26,33,57,32,56,31,55)(13,34,19,43,37,61,36,52)(15,38,42,54,30,59,41,53)(21,45,51,66,50,65,49,64)(35,60,44,63,58,68,62,67);(5,9)(6,15)(7,21)(8,26)(10,33)(11,31)(12,32)(13,35)(14,38)(16,42)(17,41)(18,30)(19,44)(20,45)(22,51)(23,49)(24,50)(25,53)(27,57)(28,55)(29,56)(34,60)(36,62)(37,58)(39,54)(40,59)(43,63)(46,66)(47,64)(48,65)(52,67)(61,68)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=fi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gi,g^b=gi,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, 300226);