Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1865 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22,26,27)(23,28,30,25)(24,32,34,29)(31,35,36,33);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23,26,30)(22,28,27,25)(24,33,34,35)(29,36,32,31);(2,4)(21,24,22,29,26,34,27,32)(23,31,25,35,30,36,28,33);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(24,34)(29,32)(31,36)(33,35);(7,15)(10,13)(14,20)(18,19)(21,25,27,23,26,28,22,30)(24,33,32,36,34,35,29,31)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=hi,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72308140 |
| maximal normal subgroup of | #72308140 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6715849);