Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25443 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,8,7)(9,10)(11,17)(12,22)(13,16)(14,19)(15,20)(18,24)(21,23);(5,7,8,6)(10,16,20,19)(11,21,18,22)(12,23)(17,24);(1,2,3,4)(9,11)(10,17)(12,19)(13,21)(14,22)(15,18)(16,23)(20,24);(9,12,14,17,15,23,13,24)(10,18,16,21,20,11,19,22)Polycyclic presentation:
2^9|a^2=h,b^2=egh,c^2=f,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216140 |
| maximal normal subgroup of | #72216140 |
| maximal quotient of | #72216072 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 274527);