Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31418 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| 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 | |
1192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2,3,4)(5,6)(7,17)(8,13)(9,26)(10,30)(11,16)(12,20)(14,28)(15,33)(18,46)(19,49)(21,42)(22,44)(23,54)(24,58)(25,39)(27,40)(29,43)(31,36)(32,41)(34,53)(35,50)(37,56)(38,60)(45,51)(47,62)(48,63)(52,68)(55,64)(57,66)(59,65)(61,67);(5,7)(6,12)(8,22)(9,27)(10,31)(11,21)(13,34)(14,40)(15,42)(16,36)(17,30)(18,25)(19,28)(20,33)(23,55)(24,48)(26,35)(29,51)(32,50)(37,64)(38,46)(39,63)(41,45)(43,49)(44,58)(47,59)(52,56)(53,60)(54,62)(57,61)(65,67)(66,68);(2,4)(5,8,11,25)(6,13,16,39)(7,18)(9,23,29,57)(10,24,33,60)(12,34)(14,37,41,65)(15,38,30,58)(17,44)(19,47)(20,48)(21,22)(26,54,43,66)(27,61)(28,56,32,59)(31,53)(35,62)(36,63)(40,64)(42,46)(45,67)(49,68)(50,52)(51,55);(2,4)(5,9,11,29)(6,14,16,41)(7,19,21,50)(8,23,25,57)(10,32,33,28)(12,35,36,49)(13,37,39,65)(15,43,30,26)(17,45,42,40)(18,47,22,52)(20,51,31,27)(24,59,60,56)(34,62,63,68)(38,66,58,54)(44,67,46,64)(48,55,53,61)Polycyclic presentation:
2^9|a^2=h,c^2=fhi,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296221);