Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(9,10,14,16)(11,18,20,24)(12,21,17,13)(15,22,23,19);(5,6)(7,8)(9,11,14,20)(10,15,16,23)(12,18,17,24)(13,22,21,19);(2,4)(6,8);(2,4)(9,12,13,10,14,17,21,16)(11,15,19,18,20,23,22,24);(10,16)(12,17)(15,23)(18,24)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,f^a=fi,c^b=ch,d^b=df,f^b=fi,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72315330, #72315342, #72315344 |
| maximal normal subgroup of | #72315330, #72315342, #72315344 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409338);