Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200022 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2655 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,28,20,25)(21,26,22,31)(24,30,32,29);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16);(11,15)(13,16)(17,19,23,20)(18,24,27,32)(21,29,22,30)(25,31,28,26);(3,8)(5,7)(10,14)(11,15)(17,20,23,19)(18,25,27,28)(21,29,22,30)(24,31,32,26);(2,6)(5,7)(11,15)(13,16)(17,21,23,22)(18,26,27,31)(19,29,20,30)(24,25,32,28)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=i,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72285141 |
| maximal normal subgroup of | #72285141 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7107548);