Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3274 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2893 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)(5,6,8,7)(9,10,14,17)(11,19,21,24)(12,18,13,16)(15,23,20,22);(9,11,14,21)(10,15,17,20)(12,22,13,23)(16,19,18,24);(2,4)(9,10)(11,20)(12,16)(13,18)(14,17)(15,21)(19,23)(22,24);(9,12)(10,16)(13,14)(17,18)(19,24)(22,23);(5,7,8,6)(10,17)(11,21)(16,18)(22,23)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,f^a=fi,c^b=ci,d^b=df,e^b=ei,f^b=fi,e^c=ei,f^d=fi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72214822, #72311727, #72311796 |
| maximal normal subgroup of | #72214822, #72311727, #72311796 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6291052);