Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24295 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#578 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,11,16,12,17,10,18)(7,19,22,13,24,38,23,37)(8,25,28,41,30,14,29,42)(9,31,32,15,34,44,33,43)(20,39,48,35,49,45,26,54)(21,36,51,62,52,63,50,46)(27,56,58,40,59,60,57,53)(47,66,67,64,68,61,55,65);(1,2,3,4)(5,7,10,23,12,24,11,22)(6,13,18,19,17,37,16,38)(8,26,29,49,30,48,28,20)(9,21,33,50,34,52,32,51)(14,39,41,54,25,45,42,35)(15,36,31,46,43,63,44,62)(27,55,57,68,59,67,58,47)(40,64,56,66,53,65,60,61);(2,4)(5,8,12,30)(6,14,17,25)(7,20,24,49)(9,27,34,59)(10,28,11,29)(13,35,37,54)(15,40,43,53)(16,41,18,42)(19,45,38,39)(21,47,52,68)(22,26,23,48)(31,60,44,56)(32,57,33,58)(36,61,63,66)(46,65,62,64)(50,67,51,55);(5,9)(6,15)(7,21)(8,27)(10,33)(11,32)(12,34)(13,36)(14,40)(16,44)(17,43)(18,31)(19,46)(20,47)(22,51)(23,50)(24,52)(25,53)(26,55)(28,58)(29,57)(30,59)(35,61)(37,63)(38,62)(39,64)(41,56)(42,60)(45,65)(48,67)(49,68)(54,66)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^c=gi,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268915);