Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021375 |
| 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 | |
1666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,9,13)(4,11,12,7)(10,16,15,14)(17,18,22,28,24,29,23,40)(19,30,32,39,33,21,27,25)(20,34,36,43,38,47,37,26)(31,41,45,44,46,48,42,35);(3,9)(4,12)(6,13)(7,11)(17,19)(18,25)(20,31)(21,28)(22,27)(23,32)(24,33)(26,41)(29,39)(30,40)(34,35)(36,42)(37,45)(38,46)(43,48)(44,47);(1,3,5,9)(2,6,8,13)(4,10,12,15)(7,14,11,16)(17,20,23,37,24,38,22,36)(18,26,28,34,29,43,40,47)(19,31,32,45,33,46,27,42)(21,35,39,48,30,44,25,41);(3,9)(6,13)(10,15)(14,16)(17,21,23,39,24,30,22,25)(18,27,40,33,29,32,28,19)(20,35,37,48,38,44,36,41)(26,42,47,46,43,45,34,31);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(17,22,24,23)(18,28,29,40)(19,27,33,32)(20,36,38,37)(21,39,30,25)(26,34,43,47)(31,42,46,45)(35,48,44,41)Polycyclic presentation:
2^9|a^2=fh,c^2=gh,d^2=fi,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,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7104026);