Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #185261 |
| 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 | |
2990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10)(11,17,19,14)(12,15)(13,16)(18,21,24,23)(20,22);(1,2)(3,8)(4,6)(5,7)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23);(3,7)(5,8)(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24);(10,16)(14,17)(15,22)(21,23)Polycyclic presentation:
2^9|a^2=g,e^2=i,b^a=be,c^a=cf,d^a=dhi,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 | #72232536, #72232543, #72232613, … |
| maximal normal subgroup of | #72232536, #72232543, #72232613, … |
| maximal quotient of | #72258036, #72338820, #72341016, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92457);