Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #457102 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1289 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)(5,6,10,17,12,18,11,16)(7,19,22,13,24,38,23,39)(8,25,28,43,30,42,29,14)(9,31,33,46,35,45,34,15)(20,48,50,63,51,36,26,40)(21,37,53,61,55,65,54,47)(27,58,32,44,60,41,59,56)(49,64,52,62,68,66,57,67);(5,7,12,24)(6,13)(8,26,30,50)(9,21,35,55)(10,22,11,23)(14,40)(15,37)(16,19)(17,38)(18,39)(20,29,51,28)(25,48)(27,57,60,52)(31,61)(32,49,59,68)(33,53,34,54)(36,42)(41,66)(43,63)(44,62)(45,47)(46,65)(56,67)(58,64);(5,8,12,30)(6,14,18,43)(7,20,24,51)(9,32,35,59)(10,29,11,28)(13,36,39,48)(15,44,46,56)(16,25,17,42)(19,40,38,63)(21,52,55,57)(22,26,23,50)(27,34,60,33)(31,58,45,41)(37,64,65,66)(47,62,61,67)(49,54,68,53);(2,4)(5,9,11,34,12,35,10,33)(6,15,17,31,18,46,16,45)(7,21,23,54,24,55,22,53)(8,27,29,59,30,60,28,32)(13,37,38,61,39,65,19,47)(14,41,25,56,43,58,42,44)(20,49,26,57,51,68,50,52)(36,62,40,66,48,67,63,64)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=i,d^2=ei,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364453);