Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62253 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60734 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C22⋊C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,17,14,18,12,16)(11,15,21,22,19,23,20,24);(1,3)(2,5)(4,8)(6,7)(9,11,13,21,14,19,12,20)(10,15,17,22,18,23,16,24);(10,16,18,17)(11,19)(15,22,23,24)(20,21);(3,8)(5,7);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df104 conjugacy classes — view the full character table.
| maximal subgroup of | #72219559, #72219565, #72224575, … |
| maximal normal subgroup of | #72219559, #72219565, #72224575, … |
| maximal quotient of | #72310142, C2×#6396161, #72317290, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6303357);