Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455564 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1112 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)(5,6,11,16)(7,17,20,12)(8,22,26,40)(9,14,28,39)(10,29,31,15)(13,30,37,25)(18,43,23,52)(19,41,47,33)(21,34,49,42)(24,51,54,62)(27,55,57,38)(32,45,35,50)(36,56,61,53)(44,64,46,66)(48,60,68,63)(58,65,59,67);(5,7,10,21,11,20,31,49)(6,12,15,34,16,17,29,42)(8,23,25,50,26,18,30,45)(9,19,27,48,28,47,57,68)(13,35,22,43,37,32,40,52)(14,33,38,60,39,41,55,63)(24,46,53,67,54,44,56,65)(36,59,51,64,61,58,62,66);(5,8,11,26)(6,13,16,37)(7,18,20,23)(9,24,28,54)(10,30,31,25)(12,32,17,35)(14,36,39,61)(15,40,29,22)(19,46,47,44)(21,50,49,45)(27,56,57,53)(33,59,41,58)(34,43,42,52)(38,62,55,51)(48,65,68,67)(60,66,63,64);(2,4)(5,9)(6,14)(7,19)(8,24)(10,27)(11,28)(12,33)(13,36)(15,38)(16,39)(17,41)(18,44)(20,47)(21,48)(22,51)(23,46)(25,53)(26,54)(29,55)(30,56)(31,57)(32,58)(34,60)(35,59)(37,61)(40,62)(42,63)(43,64)(45,65)(49,68)(50,67)(52,66)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,e^b=ei,g^b=gi,d^c=de,f^c=fi,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362763);