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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3182 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)(9,10)(11,12)(13,16)(14,18)(15,22)(17,24)(19,21)(20,23);(1,2,4,3)(5,6)(7,8)(9,11,14,19)(10,12,18,21)(13,15,24,20)(16,22,17,23);(9,12)(10,15)(11,16)(13,23)(14,21)(17,19)(18,20)(22,24);(6,8)(10,16)(11,15)(13,24)(17,18)(19,20)(22,23);(1,3,4,2)(11,19)(12,21)(15,20)(22,23)Polycyclic presentation:
2^9|b^2=fi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72251222, #72256861, #72256867, … |
| maximal normal subgroup of | #72251222, #72256861, #72256867, … |
| maximal quotient of | #72264717, #72265672, #72315893, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426463);