Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421913 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15299 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23445 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,14,8,11,7,3)(4,12,17,9,19,27,18,15)(5,20,22,21,23,10,24,13)(16,26,29,28,30,32,31,25);(1,3,6,11,8,14,7,2)(4,15,17,27,19,9,18,12)(5,21,22,20,23,13,24,10)(16,25,29,32,30,28,31,26);(1,4)(2,9,11,15)(3,12,14,27)(5,16)(6,17)(7,18)(8,19)(10,25,20,28)(13,26,21,32)(22,29)(23,30)(24,31);(1,5)(2,10)(3,13)(4,16)(6,24)(7,22)(8,23)(9,25)(11,20)(12,26)(14,21)(15,28)(17,31)(18,29)(19,30)(27,32)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,h^c=hi,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72285951, #72286235, #72286242 |
| maximal normal subgroup of | #72285951, #72286235, #72286242 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329109);