Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25798 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| 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 | |
1662 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)(15,27)(18,41)(19,45)(21,40)(22,38)(23,51)(24,54)(25,36)(29,55)(31,50)(32,48)(33,46)(34,52)(35,42)(37,56)(39,47)(43,57)(44,58)(49,65)(53,68)(59,63)(60,67)(61,66)(62,64);(5,7,10,20,11,21,27,46)(6,12,15,17,16,33,30,40)(8,22,24,50,25,18,42,44)(9,19,28,47,29,48,56,66)(13,31,35,38,36,58,54,41)(14,32,26,39,37,45,55,61)(23,49,52,67,53,43,63,64)(34,57,51,60,59,65,68,62);(2,4)(5,8,11,25)(6,13,16,36)(7,18)(9,23,29,53)(10,24,27,42)(12,31)(14,34,37,59)(15,35,30,54)(17,38)(19,43)(20,44)(21,22)(26,51,55,68)(28,52,56,63)(32,57)(33,58)(39,60)(40,41)(45,65)(46,50)(47,64)(48,49)(61,62)(66,67);(5,9,27,56,11,29,10,28)(6,14,30,55,16,37,15,26)(7,19,46,66,21,48,20,47)(8,23,42,63,25,53,24,52)(12,32,40,61,33,45,17,39)(13,34,54,68,36,59,35,51)(18,43,50,67,22,49,44,64)(31,57,41,62,58,65,38,60)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=fhi,d^2=ei,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281624);