Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454375 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6,8,9)(7,10,11,12)(13,14)(15,17)(16,18)(19,20)(21,22,24,31,25,30,26,29)(23,32,33,28,34,27,35,36);(6,9)(10,12)(13,15,16,19)(14,17,18,20)(21,23,26,35,25,34,24,33)(22,27,29,28,30,32,31,36);(2,4)(5,7)(6,10)(8,11)(9,12)(21,23,25,34)(22,28,30,36)(24,35,26,33)(27,31,32,29);(7,11)(10,12)(14,18)(17,20)(21,24,25,26)(22,29,30,31)(23,33,34,35)(27,28,32,36)Polycyclic presentation:
2^9|a^2=ef,b^2=gh,c^2=i,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378988);