Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518952 |
| 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 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1334 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)(9,10,12,14)(11,13,15,16)(17,18)(19,22)(20,26)(21,30)(23,27)(24,28)(25,32)(29,31);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,24,27)(18,22,28,23)(20,30,31,25)(21,29,32,26);(1,3,4,7)(2,6,5,8)(17,20,23,21,24,31,22,32)(18,25,19,29,28,30,27,26);(11,15)(13,16)(17,21,23,31,24,32,22,20)(18,26,19,25,28,29,27,30);(2,5)(6,8)(20,31)(21,32)(25,30)(26,29)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=fgi,d^2=gi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7499356);