Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514060 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | D4×#120 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3199 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18,21,20)(19,22,24,23);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,22)(20,23)(21,24);(10,14)(11,15);(10,14)(13,16)(18,20)(19,23)(22,24);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=gi,b^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72282345, #72282375, #72282515, … |
| maximal normal subgroup of | #72282345, #72282375, #72282515, … |
| maximal quotient of | #72305671, #72305748, #72306000, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7430536);