Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393940 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1224 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)(7,10)(8,9)(11,12)(13,14)(15,19)(16,17)(18,20)(21,22)(23,26)(24,27)(25,28)(29,36)(30,34)(31,33)(32,35);(6,9)(10,12)(21,23)(22,26)(24,30)(25,29)(27,34)(28,36)(31,32)(33,35);(2,4)(5,7,8,11)(6,10,9,12)(14,17)(18,19)(22,27)(23,29)(25,32)(26,33)(30,31)(34,36);(13,15,16,20)(14,18,17,19)(22,27)(26,34)(28,35)(33,36)Polycyclic presentation:
2^9|a^2=fh,c^2=e,d^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72264436, #72264500, #72264579, … |
| maximal normal subgroup of | #72264436, #72264500, #72264579, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 322279);