Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator, socle | C24 | C23×C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(9,10)(11,15)(12,16)(13,14)(17,18,23,19,22,25,21,27)(20,24,28,26,29,32,30,31);(5,6,7,8)(10,14)(15,16)(17,19,23,25,22,27,21,18)(20,26,28,32,29,31,30,24);(1,2,3,4)(6,8)(9,11,13,12)(10,15,14,16)(17,20,22,29)(18,24,25,32)(19,26,27,31)(21,30,23,28);(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,19,25,27)(20,28,29,30)(24,31,32,26)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=ef,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72239336, #72239363, #72239405, … |
| maximal normal subgroup of | #72239336, #72239363, #72239405, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 274582);