Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996395 |
| 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 |
| prime radical | #61424 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(4,12)(7,13)(11,16)(14,15)(17,18,22,27,23,28,21,26)(19,29,31,39,33,44,32,24)(20,25,36,42,37,43,35,41)(30,40,34,47,46,48,45,38);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19)(18,24)(20,34)(21,31)(22,32)(23,33)(25,40)(26,29)(27,44)(28,39)(30,36)(35,46)(37,45)(38,42)(41,47)(43,48);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,18)(19,24)(20,25)(21,26)(22,27)(23,28)(29,31)(30,38)(32,44)(33,39)(34,40)(35,41)(36,42)(37,43)(45,48)(46,47);(3,10)(6,9)(11,16)(14,15)(17,20,22,36,23,37,21,35)(18,25,27,42,28,43,26,41)(19,30,31,34,33,46,32,45)(24,38,29,40,39,47,44,48);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,37,36)(24,29,39,44)(25,41,43,42)(30,34,46,45)(38,40,47,48)Polycyclic presentation:
2^9|a^2=f,b^2=h,d^2=f,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7088161);