Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #381470 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14299 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23402 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,9,7,12,5,11)(3,13,15,8,17,23,16,22)(4,10,19,24,20,25,18,26)(14,27,28,21,29,32,30,31);(1,3)(2,8)(4,14)(5,16)(6,15)(7,17)(9,23)(10,21)(11,13)(12,22)(18,30)(19,28)(20,29)(24,32)(25,31)(26,27);(2,9)(4,18)(5,6)(8,13)(10,25)(11,12)(14,28)(15,16)(19,20)(21,31)(22,23)(29,30);(1,4,6,19,7,20,5,18)(2,10,9,24,12,25,11,26)(3,14,15,28,17,29,16,30)(8,21,23,32,22,31,13,27)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,e^b=ei,d^c=de,e^c=ei,f^c=fi,h^c=hi,f^d=fi,g^d=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72251777, #72251786, #72251801, … |
| maximal normal subgroup of | #72251777, #72251786, #72251801, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288668);