Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884937 |
| 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 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1839 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(17,18,22,30,21,25,19,23)(20,27,26,24,29,31,28,32);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,22)(18,23,25,30)(20,26,29,28)(24,31,32,27);(1,3,4,7)(2,6,5,8)(10,13)(11,15)(17,20)(18,24)(19,26)(21,29)(22,28)(23,31)(25,32)(27,30);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(18,25)(23,30)(24,32)(27,31);(10,13)(14,16)(17,19,21,22)(18,23,25,30)(20,26,29,28)(24,31,32,27)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=f,d^2=fh,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,d^c=dfh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6843964);