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 | |
3037 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)(11,13)(12,20)(14,18)(15,16)(17,21)(19,22)(23,24);(1,2,4,3)(5,6)(7,8)(9,11)(10,15)(12,17)(13,18)(14,16)(19,21)(20,23)(22,24);(10,16)(11,18)(13,15)(17,20)(19,24)(21,23);(6,8)(9,12,14,22)(10,17,18,24)(11,19,16,20)(13,21,15,23);(1,3,4,2)(9,13,14,15)(10,16,18,11)(12,21,22,23)(17,20,24,19)Polycyclic presentation:
2^9|b^2=fi,d^2=i,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 | #72263839, #72263846, #72263863, … |
| maximal normal subgroup of | #72263839, #72263846, #72263863, … |
| maximal quotient of | #72264516, #72315958, #72315966 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426465);