Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455832 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16270 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23400 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1139 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)(3,12)(4,16)(5,21)(6,10)(7,11)(8,14)(9,20)(13,26)(15,24)(17,32)(18,27)(19,28)(22,25)(23,30)(29,31);(1,3)(2,8)(4,17,20,30)(5,22)(6,14)(7,15)(9,26,16,29)(10,24)(11,25)(12,21)(13,18,31,19)(23,28,32,27);(1,4,5,19,7,20,6,18)(2,9,10,28,11,16,21,27)(3,13,14,17,15,31,22,30)(8,23,12,29,25,32,24,26);(4,18)(5,6)(9,27)(10,21)(12,24)(13,30)(14,22)(16,28)(17,31)(19,20)(23,26)(29,32)Polycyclic presentation:
2^9|b^2=f,c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237054, #72237064, #72237073, … |
| maximal normal subgroup of | #72237054, #72237064, #72237073, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363028);