Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6342800 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#123 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3072 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,23)(20,30,32,24)(21,25)(22,26)(27,39,41,34)(28,35)(29,36)(31,43,44,37)(33,38)(40,47,48,45)(42,46);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,23)(20,27)(21,28)(22,29)(24,34)(25,35)(26,36)(30,39)(31,40)(32,41)(33,42)(37,45)(38,46)(43,47)(44,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,24)(19,27)(21,31)(22,32)(23,34)(25,37)(26,30)(28,40)(29,41)(33,44)(35,45)(36,39)(38,43)(42,48)(46,47);(3,11)(6,9)(10,16)(13,15)(17,21,22,33)(18,25,26,38)(19,28,29,42)(20,31,32,44)(23,35,36,46)(24,37,30,43)(27,40,41,48)(34,45,39,47);(2,8)(6,9)(7,14)(13,15)(19,29)(23,36)(27,41)(28,42)(34,39)(35,46)(40,48)(45,47)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6293308);