Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454039 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60371 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1823 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,5)(3,6,7,8)(9,10,13,17,15,12,14,18)(11,16,21,19,22,20,23,24);(2,5)(6,8)(9,11)(10,16)(12,20)(13,21)(14,23)(15,22)(17,19)(18,24);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,19)(13,17)(14,18)(16,21)(20,23)(22,24);(3,7)(6,8)(9,12)(10,15)(11,20)(13,18)(14,17)(16,22)(19,23)(21,24)Polycyclic presentation:
2^9|a^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72227999 |
| maximal normal subgroup of | #72227999 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361829);