Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| 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 | |
1168 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)(6,9)(10,12)(13,14)(15,17)(16,18)(19,20)(21,22,27,32,25,30,26,31)(23,33,35,36,24,28,34,29);(5,6,8,9)(7,10,11,12)(13,15,16,19)(14,17,18,20)(21,23,27,35,25,24,26,34)(22,28,32,29,30,33,31,36);(5,7,8,11)(6,10,9,12)(14,18)(17,20)(21,24)(22,29)(23,25)(26,34)(27,35)(28,32)(30,36)(31,33);(2,4)(7,11)(10,12)(22,30)(23,24)(31,32)(34,35)Polycyclic presentation:
2^9|a^2=g,b^2=ef,c^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367894);