Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #482357 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1231 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)(3,5)(4,6)(7,8)(9,10)(11,17)(12,20,21,15)(13,16)(14,18,22,24)(19,23);(2,5)(3,8)(6,7)(9,11,13,19)(10,14,16,22)(12,18,21,24)(15,17,20,23);(2,6)(5,7);(2,6)(5,7)(9,12)(10,15)(11,18)(13,21)(14,17)(16,20)(19,24)(22,23)Polycyclic presentation:
2^9|a^2=f,b^2=i,e^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,f^d=fi,f^e=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72240244, #72240245, #72240250, … |
| maximal normal subgroup of | #72240244, #72240245, #72240250, … |
| maximal quotient of | #72240339, #72293143 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389554);