Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4⋊C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1068 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(10,14,16,20)(11,17)(15,22,23,18)(19,24);(5,6,7,8)(9,10)(11,18)(12,20)(13,16)(14,21)(15,19)(17,22)(23,24);(1,2,3,4);(9,11,12,19,13,17,21,24)(10,15,14,18,16,23,20,22)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=di,d^b=de,e^b=ei,g^b=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72309847, #72309852, D4×#1773, … |
| maximal normal subgroup of | #72309847, #72309852, D4×#1773, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263554);