Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5845 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,13,16)(11,14,19,23)(12,15,20,24)(17,21,18,22);(2,4)(9,11)(10,14)(12,17)(13,19)(15,21)(16,23)(18,20)(22,24);(6,8)(11,17)(12,20)(14,21)(15,24)(18,19)(22,23);(9,11)(10,14)(12,17)(13,19)(15,21)(16,23)(18,20)(22,24);(11,17)(12,20)(14,21)(15,24)(18,19)(22,23)Polycyclic presentation:
2^9|a^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72238725, #72249378, #72249459, … |
| maximal normal subgroup of | #72238725, #72249378, #72249459, … |
| maximal quotient of | #72239785, #72265282, #72306618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480667);