Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13343 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10)(11,17,19,24)(12,20,22,15)(13,16)(14,18)(21,23);(5,6)(7,8)(9,11,13,19)(10,14,16,23)(12,21,22,18)(15,24,20,17);(2,4)(6,8)(15,20)(17,24)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=di,e^b=ehi,g^b=ghi,d^c=dh,e^d=ei,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72234548, #72234555, #72234569, … |
| maximal normal subgroup of | #72234548, #72234555, #72234569, … |
| maximal quotient of | #72234550, #72250914, #72284597, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55022);