Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394048 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(10,15)(11,18)(12,13)(17,20)(19,21)(22,23);(5,6)(7,8)(9,10)(11,16)(12,20)(13,15)(14,17)(18,23)(19,24)(21,22);(2,4)(6,8);(9,11,14,19)(10,16,17,24)(12,18,13,21)(15,22,20,23)Polycyclic presentation:
2^9|d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,h^d=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72239576, C2×#394095 |
| maximal normal subgroup of | #72239576, C2×#394095 |
| maximal quotient of | #72249987, #72261528, #72261542, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301291);