Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455690 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2825 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16379 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
871 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,26,8,13,6,12)(3,14,16,28,17,29,20,9)(4,18,15,31,22,10,21,27)(5,23,19,32,25,30,24,11);(1,3)(2,9,13,28)(4,19)(5,15)(6,16)(7,20)(8,17)(10,30,18,23)(11,27,32,31)(12,14,26,29)(21,25)(22,24);(1,4)(2,10)(3,5)(6,15)(7,21)(8,22)(9,11)(12,27)(13,18)(14,30)(16,24)(17,25)(19,20)(23,29)(26,31)(28,32);(1,5)(2,11)(3,15)(4,20)(6,24)(7,19)(8,25)(9,27)(10,29)(12,30)(13,32)(14,18)(16,22)(17,21)(23,26)(28,31)Polycyclic presentation:
2^9|a^2=fg,b^2=f,e^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,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,e^d=ei,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72318910, #72318935, #72318969, … |
| maximal normal subgroup of | #72318910, #72318935, #72318969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368293);