Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667748 |
| 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 | |
1671 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)(3,10)(4,13)(5,8)(6,9)(7,12)(11,15)(14,16)(17,18)(19,29)(20,27)(21,26)(22,25)(23,28)(24,32)(30,40)(31,45)(33,39)(34,48)(35,42)(36,41)(37,44)(38,47)(43,46);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,10)(13,16)(17,19)(18,24)(20,34)(21,32)(22,33)(23,30)(25,40)(26,29)(27,43)(28,39)(31,37)(35,46)(36,47)(38,44)(41,45)(42,48);(18,25)(19,30)(22,23)(26,28)(27,41)(31,46)(32,33)(34,47)(36,37)(39,40)(42,44)(43,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(18,26)(24,29)(25,28)(27,42)(38,45)(39,40)(41,44)(43,48);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,20)(18,27)(19,31)(21,35)(22,36)(23,37)(24,38)(25,41)(26,42)(28,44)(29,45)(30,46)(32,47)(33,34)(39,48)(40,43)Polycyclic presentation:
2^9|d^2=fi,e^2=fi,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,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6582729);