Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996799 |
| 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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1611 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)(3,6,10,14)(4,7,12,16)(9,13,11,15)(17,18,24,30,23,28,22,29)(19,31,35,25,33,45,34,43)(20,36,39,26,40,48,38,46)(21,27,32,42,41,47,37,44);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19)(18,25)(20,37)(21,38)(22,35)(23,33)(24,34)(26,44)(27,36)(28,43)(29,45)(30,31)(32,40)(39,41)(42,46)(47,48);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(17,20)(18,26)(19,32)(21,34)(22,38)(23,40)(24,39)(25,42)(27,31)(28,46)(29,36)(30,48)(33,37)(35,41)(43,44)(45,47);(4,12)(7,16)(9,11)(13,15)(17,21,24,32,23,41,22,37)(18,27,30,42,28,47,29,44)(19,20,35,39,33,40,34,38)(25,26,45,48,43,46,31,36);(2,8)(3,10)(7,16)(9,11)(18,28)(19,33)(21,41)(26,46)(29,30)(32,37)(34,35)(36,48)Polycyclic presentation:
2^9|a^2=fh,c^2=hi,d^2=fi,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=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7012005);