Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418498 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2673 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15320 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,13,8,26)(3,14,17,28,19,27,18,9)(4,20,22,5,23,10,11,25)(15,30,21,32,29,16,31,24);(1,3)(2,9)(4,21)(5,24)(6,18)(7,19)(8,17)(10,16)(11,29)(12,27)(13,28)(14,26)(15,22)(20,30)(23,31)(25,32);(1,4,8,11,7,23,6,22)(2,10,26,5,13,20,12,25)(3,15,17,21,19,29,18,31)(9,24,14,30,28,32,27,16);(1,5,6,10,7,25,8,20)(2,11,12,4,13,22,26,23)(3,16,17,24,19,30,18,32)(9,21,14,29,28,31,27,15)Polycyclic presentation:
2^9|a^2=e,c^2=ef,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72339026, #72339034, #72339041 |
| maximal normal subgroup of | #72339026, #72339034, #72339041 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 331514);