Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499831 |
| 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 | #51749 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1109 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 1109 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(12,23)(15,29)(16,24)(20,34)(21,27)(25,37)(31,39)(33,35);(2,5)(6,8)(9,10,14,17)(11,19,22,32)(12,24,23,16)(13,26,28,18)(15,27,29,21)(20,35,34,33)(25,31,37,39)(30,36,38,40);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,20)(13,27)(14,22)(16,30)(17,29)(18,32)(19,26)(21,28)(23,34)(24,38)(25,40)(31,35)(33,39)(36,37);(2,5)(6,8)(9,12,14,23)(10,16,17,24)(11,20,22,34)(13,25,28,37)(15,30,29,38)(18,31,26,39)(19,33,32,35)(21,36,27,40)Polycyclic presentation:
2^9|b^2=i,c^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,g^c=gi,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72311266, #72311281, #72311286, … |
| maximal normal subgroup of | #72311266, #72311281, #72311286, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407388);