Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1647 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)(3,6)(4,5)(7,8)(9,10)(11,17,19,14)(12,16)(13,15)(18,24,21,22)(20,23);(2,5)(6,8)(11,18)(12,20)(14,22)(16,23)(17,24)(19,21);(9,11)(10,14)(12,21)(13,19)(15,17)(16,24)(18,20)(22,23);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=h,d^2=e,g^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,g^b=gi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72226667, #72226675, #72226676, … |
| maximal normal subgroup of | #72226667, #72226675, #72226676, … |
| maximal quotient of | #72264038, #72264226, C2×#358176 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265372);