Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421911 |
| 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 | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1191 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,8,11,7,3)(4,13,16,9,17,25,22,24)(5,18,19,10,21,14,20,26)(15,28,29,32,31,23,30,27);(1,3,6,11,8,12,7,2)(4,9,16,13,17,24,22,25)(5,18,19,26,21,14,20,10)(15,28,29,27,31,23,30,32);(1,4)(2,9,11,24)(3,13,12,25)(5,15)(6,16)(7,22)(8,17)(10,23,26,28)(14,27,18,32)(19,29)(20,30)(21,31);(1,5)(2,10)(3,14)(4,15)(6,20)(7,19)(8,21)(9,23)(11,26)(12,18)(13,27)(16,30)(17,31)(22,29)(24,28)(25,32)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72237066, #72237072, #72237090, … |
| maximal normal subgroup of | #72237066, #72237072, #72237090, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329107);