Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11505 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C24 |
| socle | C22 | C22×#99 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,8,5,6,3,7)(9,10)(11,19)(12,16)(13,17,22,23)(14,18)(15,20,24,21);(9,11,14,12)(10,15,18,24)(13,21,22,20)(16,23,19,17);(2,6)(7,8);(1,3,5,4)(2,7,6,8)(9,12,14,11)(10,16,18,19)(13,21,22,20)(15,23,24,17)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=hi,e^2=i,f^2=h,b^a=be,c^a=ch,e^a=eg,g^a=gi,e^b=ei,f^b=fgi,f^e=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72338671, #72338753 |
| maximal normal subgroup of | #72338671, #72338753 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387587);