Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1660 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)(17,18)(19,26)(20,22)(21,25)(23,30)(24,27)(28,32)(29,31);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(18,20)(24,29)(25,30)(26,32);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,23,27,22,28,21,31)(18,24,25,26,20,29,30,32);(2,6)(5,7)(11,16)(13,15)(17,20,22,18)(19,24,28,29)(21,25,23,30)(26,31,32,27);(10,14)(13,15)(17,21,22,23)(18,25,20,30)(19,27,28,31)(24,32,29,26)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=g,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72308682, #72308764 |
| maximal normal subgroup of | #72308682, #72308764 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6816608);