Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #185255 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3490 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,17,19,24)(12,16)(13,15)(14,18,22,20)(21,23);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,20)(13,19)(15,22)(16,24)(17,23)(18,21);(1,2)(3,5)(4,6)(7,8);(10,15)(14,22)(16,23)(17,24)Polycyclic presentation:
2^9|a^2=g,e^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,f^b=fh,g^b=ghi,e^c=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72219034, #72231950, #72232711, … |
| maximal normal subgroup of | #72219034, #72231950, #72232711, … |
| maximal quotient of | #72231809, #72237874, #72241259, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92451);