Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| 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 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2953 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)(9,10,14,18,16,13,15,19)(11,20,23,17,21,24,12,22);(5,6)(7,8)(9,11)(10,17)(12,14)(13,22)(15,23)(16,21)(18,20)(19,24);(1,2,3,4)(6,8)(9,12,14,11,16,23,15,21)(10,17,18,24,13,22,19,20);(6,8)(9,13,15,18,16,10,14,19)(11,20,12,22,21,24,23,17);(9,14,16,15)(10,18,13,19)(11,12,21,23)(17,20,22,24)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72315519, #72315550 |
| maximal normal subgroup of | #72315519, #72315550 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351571);