Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #461427 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16391 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,5,12,7,21)(3,8,15,10,14,23,20,24)(4,9,17,28,19,27,18,32)(13,29,16,26,30,22,31,25);(1,3)(2,8)(4,16)(5,14)(6,20)(7,15)(9,25)(10,21)(11,24)(12,23)(13,17)(18,30)(19,31)(22,28)(26,27)(29,32);(1,4)(2,9)(3,13,14,30)(5,19)(6,17)(7,18)(8,22,23,29)(10,25,24,26)(11,28)(12,27)(15,16,20,31)(21,32);(1,3)(2,10)(4,13)(5,14)(6,15)(7,20)(8,21)(9,26)(11,23)(12,24)(16,17)(18,31)(19,30)(22,28)(25,27)(29,32)Polycyclic presentation:
2^9|a^2=g,c^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,g^b=gi,d^c=de,e^c=ei,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251787, #72251790, #72251800, … |
| maximal normal subgroup of | #72251787, #72251790, #72251800, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368623);