Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3077 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(21,22,26,31,27,32,25,30)(23,33,34,36,24,29,35,28);(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23,27,24)(22,28,32,36)(25,34,26,35)(29,30,33,31);(2,4)(6,10)(11,17)(12,18)(19,20)(21,24,27,23)(22,29,32,33)(25,34,26,35)(28,30,36,31);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(21,25,27,26)(22,30,32,31)(23,34,24,35)(28,33,36,29);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=i,d^2=hi,e^2=h,f^2=i,b^a=bf,c^a=cg,c^b=ch,f^b=fi,d^c=di,e^d=eh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72284454, #72284723 |
| maximal normal subgroup of | #72284454, #72284723 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6451343);