Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3041 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1593 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(5,6)(7,8)(10,15,18,16)(11,19,21,20)(17,24)(22,23);(1,2,4,3)(5,6)(7,8)(9,10,12,16,14,18,13,15)(11,17,19,23,21,24,20,22);(6,8)(10,16)(12,13)(15,18)(17,23)(19,20)(22,24);(1,3,4,2)(9,11)(10,17)(12,19)(13,20)(14,21)(15,22)(16,23)(18,24)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=efg,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,g^c=gi,e^d=ei,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72226917, #72226931, #72226937 |
| maximal normal subgroup of | #72226917, #72226931, #72226937 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 279904);