Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 |
| prime radical | C2×#3155 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2887 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,8,7)(10,15)(16,18)(17,23)(19,24);(6,7)(9,10,13,15)(11,19,21,24)(12,18,14,16)(17,22,23,20);(1,2,3,4)(5,6,8,7)(10,16)(12,14)(15,18)(17,19)(20,22)(23,24);(5,7,8,6)(9,11,13,21)(10,17,15,23)(12,20,14,22)(16,19,18,24);(9,12,13,14)(10,16,15,18)(11,20,21,22)(17,19,23,24)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fgi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72342360, #72342369, #72342373 |
| maximal normal subgroup of | #72342360, #72342369, #72342373 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351404);