Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518944 |
| 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 | |
2999 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(6,7)(9,10,16,11)(12,19,22,17)(13,14,15,18)(20,21,23,24);(2,4)(6,7)(9,11,15,18,16,10,13,14)(12,17,20,21,22,19,23,24);(5,6,8,7)(9,12,16,22)(10,17,11,19)(13,23,15,20)(14,21,18,24);(1,2,3,4)(5,7,8,6)(9,13,16,15)(10,18,11,14)(12,20,22,23)(17,21,19,24);(9,14,15,11,16,18,13,10)(12,21,20,19,22,24,23,17)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=hi,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72261157, #72261248, #72261272, … |
| maximal normal subgroup of | #72261157, #72261248, #72261272, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521243);