Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480319 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | #20723 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ #97 ◁ #2875 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
753 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12,15,8)(4,9)(5,10)(6,11)(7,14,21,29)(13,27,16,22)(17,24)(18,25)(19,26)(20,28,23,30)(31,32);(1,3)(2,7,11,21)(4,16)(5,14)(6,15)(8,10,12,26)(9,23,25,20)(13,18)(17,30)(19,29)(22,32,27,24)(28,31);(2,8)(7,10)(9,22)(11,12)(20,24)(21,26)(23,32)(25,27);(1,4,5,17,6,18,19,31)(2,9,10,24,11,25,26,32)(3,13,14,28,15,16,29,30)(7,20,12,27,21,23,8,22)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=e,e^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,g^b=gi,g^c=gh,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231615, #72231703, #72231704, … |
| maximal normal subgroup of | #72231615, #72231703, #72231704, … |
| maximal quotient of | C2×#494016, #72264437, #72264443, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401212);