Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| 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 | |
1158 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)(5,6,8,10)(7,11,12,9)(13,14,16,18)(15,17,19,20)(21,22,25,28)(23,26,31,34)(24,27,30,33)(29,32,35,36);(5,7,8,12)(6,9,10,11)(13,15,16,19)(14,17,18,20)(23,29)(24,30)(26,32)(27,33)(31,35)(34,36);(14,18)(17,20)(21,23,30,35,25,31,24,29)(22,26,33,36,28,34,27,32);(2,4)(7,12)(9,11)Polycyclic presentation:
2^9|a^2=efi,b^2=ef,c^2=gi,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72308317, #72308323, #72308387, … |
| maximal normal subgroup of | #72308317, #72308323, #72308387, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 351184);