Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455686 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16268 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23407 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1115 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,12)(4,17)(5,20)(8,13)(9,25)(14,30)(15,16)(18,19)(21,24)(22,27)(23,26);(1,2,5,11,6,7,20,10)(3,13,12,21,16,24,15,8)(4,9,18,25,19,22,17,27)(14,29,28,26,30,32,31,23);(1,3)(2,8)(4,14)(5,15)(6,16)(7,21)(9,26)(10,13)(11,24)(12,20)(17,28)(18,31)(19,30)(22,23)(25,29)(27,32);(1,4)(2,9)(3,14)(5,18)(6,19)(7,22)(8,23)(10,27)(11,25)(12,28)(13,29)(15,31)(16,30)(17,20)(21,26)(24,32)Polycyclic presentation:
2^9|b^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,d^c=de,f^c=fi,g^c=gi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237069, #72237072, #72237086, … |
| maximal normal subgroup of | #72237069, #72237072, #72237086, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362882);