Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324468 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1673 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(10,13)(11,14)(15,16)(17,18,22,26)(19,27,29,23)(20,24,21,25)(28,32,30,31);(1,2)(3,7)(4,6)(5,8)(17,19,22,29)(18,23,26,27)(20,28,21,30)(24,31,25,32);(1,3)(2,5)(4,8)(6,7)(9,10,14,16,12,15,11,13)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,31,27,32);(2,6)(3,8)(17,21,22,20)(18,25,26,24)(19,28,29,30)(23,31,27,32);(2,6)(5,7)(10,15)(13,16)(18,26)(23,27)(24,25)(31,32)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fgi,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,e^c=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72282984, #72283727, #72283966, … |
| maximal normal subgroup of | #72282984, #72283727, #72283966, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7238356);