Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499795 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×B(4,𝔽2) | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1021 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8)(3,7,4,6)(9,10,12,14)(11,17,18,24)(13,16,20,22)(15,23,21,19);(1,3,5,4)(2,6,8,7)(10,14)(13,20)(15,21)(17,24);(9,11)(10,15)(12,18)(13,19)(14,21)(16,17)(20,23)(22,24);(1,4,5,3)(2,7,8,6)Polycyclic presentation:
2^9|a^2=e,b^2=ei,d^2=ei,b^a=be,c^a=cf,d^a=dei,g^a=ghi,c^b=cg,f^b=fh,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72278314, #72278320, #72278399, … |
| maximal normal subgroup of | #72278314, #72278320, #72278399, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408148);