Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500689 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3397 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(10,14)(11,17)(12,19)(16,20)(18,21)(22,24);(1,2)(3,4)(5,6)(7,8);(6,8)(9,10)(11,15)(12,20)(13,16)(14,19)(17,22)(18,23)(21,24);(2,4)(9,11,13,18)(10,15,16,23)(12,21,19,17)(14,22,20,24);(10,16)(14,20)(15,23)(22,24)Polycyclic presentation:
2^9|a^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,e^c=ei,g^c=gi,g^d=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72307380, #72307394, #72307451, … |
| maximal normal subgroup of | #72307380, #72307394, #72307451, … |
| maximal quotient of | #72315679, #72315745 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408159);