Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #34738 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3409 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,22)(23,24);(1,2,4,3)(5,6)(7,8)(9,11,13,19)(10,14,16,22)(12,18,20,24)(15,21,17,23);(1,3,4,2)(6,8)(10,15)(12,20)(14,21)(16,17)(18,24)(22,23);(6,8);(10,16)(14,22)(15,17)(21,23)Polycyclic presentation:
2^9|b^2=fi,c^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72282852, #72283423, #72283695, … |
| maximal normal subgroup of | #72282852, #72283423, #72283695, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345961);