Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455690 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2825 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16268 |
| prime radical | #23406 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1082 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)(3,8)(4,18)(5,22)(6,10)(7,24)(9,21)(11,12)(13,29)(14,31)(15,27)(16,28)(17,26)(19,32)(20,23)(25,30);(1,3,12,15,6,16,7,17)(2,8,24,26,10,28,11,27)(4,19,22,31,20,13,21,30)(5,14,18,32,9,25,23,29);(1,4)(2,9)(3,13)(5,10)(6,20)(7,22)(8,25)(11,23)(12,21)(14,28)(15,31)(16,19)(17,30)(18,24)(26,32)(27,29);(1,5)(2,4)(3,14)(6,9)(7,23)(8,19)(10,20)(11,21)(12,18)(13,28)(15,32)(16,25)(17,29)(22,24)(26,31)(27,30)Polycyclic presentation:
2^9|b^2=ef,f^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,g^b=gi,d^c=de,f^c=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72318876, #72318935, #72318969, … |
| maximal normal subgroup of | #72318876, #72318935, #72318969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362886);