Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2873 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(6,7)(9,10,13,11)(12,15,20,17)(14,16,21,18)(19,24,22,23);(1,2,4,3)(6,7)(10,11)(12,19)(14,21)(15,23)(17,24)(20,22);(5,6,8,7)(9,11,13,10)(12,17,20,15)(14,18,21,16)(19,23,22,24);(5,7,8,6)(9,12,13,20)(10,15,11,17)(14,22,21,19)(16,23,18,24);(1,3,4,2)(10,11)(15,17)(16,18)(23,24)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gi,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470324);