Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470231 |
| 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 |
| prime radical | #62847 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
2935 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)(3,6,9,13)(4,11,12,7)(10,16,15,14)(17,18,22,26)(19,23,29,35)(20,24,31,37)(21,25,32,38)(27,39,41,33)(28,34,42,45)(30,36,43,46)(40,47,48,44);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,19)(18,23)(20,27)(21,28)(22,29)(24,33)(25,34)(26,35)(30,40)(31,41)(32,42)(36,44)(37,39)(38,45)(43,48)(46,47);(3,9)(6,13)(10,15)(14,16)(17,20)(18,24)(19,27)(21,30)(22,31)(23,33)(25,36)(26,37)(28,40)(29,41)(32,43)(34,44)(35,39)(38,46)(42,48)(45,47);(3,9)(6,13)(10,15)(14,16)(17,21,22,32)(18,25,26,38)(19,28,29,42)(20,30,31,43)(23,34,35,45)(24,36,37,46)(27,40,41,48)(33,44,39,47);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(21,32)(25,38)(28,42)(30,43)(34,45)(36,46)(40,48)(44,47)Polycyclic presentation:
2^9|a^2=hi,d^2=i,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,g^b=gi,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, 6403472);