Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6807031 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1639 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,9,11,6)(4,12,13,7)(10,14,16,15)(17,18,23,29)(19,30,34,25)(20,35,37,46)(21,38,24,27)(22,28,40,48)(26,36,43,33)(31,47,39,45)(32,42,41,44);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,25)(20,22)(21,32)(23,34)(24,41)(26,45)(27,42)(28,46)(29,30)(31,33)(35,48)(36,39)(37,40)(38,44)(43,47);(17,20,24,33,23,37,21,36)(18,26,27,46,29,43,38,35)(19,31,32,22,34,39,41,40)(25,28,44,45,30,48,42,47);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,21,23,24)(18,27,29,38)(19,32,34,41)(20,36,37,33)(22,39,40,31)(25,42,30,44)(26,46,43,35)(28,47,48,45);(3,11)(6,9)(10,16)(14,15)(17,22,21,31,23,40,24,39)(18,28,38,47,29,48,27,45)(19,33,41,20,34,36,32,37)(25,43,42,46,30,26,44,35)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=f,e^2=hi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6716300);