Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500015 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
2841 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(17,18)(19,27)(20,31,33,24)(21,25)(22,26)(23,30)(28,39,32,37)(29,41)(34,44,46,40)(35,38)(36,42)(43,48,45,47);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19)(18,23)(20,32)(21,35)(22,30)(24,39)(25,41)(26,27)(28,33)(29,36)(31,37)(34,43)(38,42)(40,47)(44,48)(45,46);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,20,22,33)(18,24,26,31)(19,28,30,32)(21,34,36,46)(23,37,27,39)(25,40,42,44)(29,43,35,45)(38,47,41,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(20,33)(21,36)(24,31)(25,42)(28,32)(29,35)(37,39)(38,41);(2,8)(6,9)(7,16)(14,15)(17,21,22,36)(18,25,26,42)(19,29,30,35)(20,34,33,46)(23,38,27,41)(24,40,31,44)(28,43,32,45)(37,47,39,48)Polycyclic presentation:
2^9|a^2=fi,c^2=h,d^2=hi,e^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6436532);