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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1621 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)(17,18,22,27,24,20,23,34)(19,28,31,29,32,41,39,25)(21,26,37,43,36,33,38,47)(30,42,35,44,45,46,48,40);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19)(18,25)(20,29)(21,35)(22,39)(23,31)(24,32)(26,42)(27,41)(28,34)(30,37)(33,46)(36,48)(38,45)(40,43)(44,47);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20,24,18)(19,29,32,25)(21,33,36,26)(22,34,23,27)(28,31,41,39)(30,44,45,40)(35,46,48,42)(37,47,38,43);(3,10)(6,9)(11,16)(14,15)(17,21,22,37,24,36,23,38)(18,26,27,43,20,33,34,47)(19,30,31,35,32,45,39,48)(25,40,28,42,29,44,41,46);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(21,36)(26,33)(30,45)(35,48)(37,38)(40,44)(42,46)(43,47)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=i,d^2=f,e^2=h,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, 7088501);