Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508070 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #24262 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
993 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,17)(11,15,20,29)(12,22,24,37)(13,18,26,32)(16,30,31,23)(19,34,35,40)(21,28,25,33)(27,38,39,36);(2,5)(6,8)(9,11)(10,15)(12,19)(13,25)(14,20)(16,27)(17,29)(18,33)(21,26)(22,34)(23,36)(24,35)(28,32)(30,38)(31,39)(37,40);(3,7)(6,8)(9,12)(10,16)(11,19)(13,23)(14,24)(15,27)(17,31)(18,22)(20,35)(21,36)(25,38)(26,30)(28,34)(29,39)(32,37)(33,40);(1,3,4,7)(2,6,5,8)(10,17)(11,20)(16,31)(18,32)(19,35)(21,25)(22,37)(36,38)Polycyclic presentation:
2^9|a^2=i,d^2=e,b^a=be,c^a=cf,d^a=di,g^a=ghi,c^b=cg,d^b=di,f^b=fh,d^c=de,h^c=hi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72317734, #72317756, #72317851, … |
| maximal normal subgroup of | #72317734, #72317756, #72317851, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415396);