Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| 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 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1698 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,20)(19,23,27,26)(22,24,30,29)(25,32,28,31);(1,3,4,8)(2,5,6,7)(17,19)(18,23)(20,26)(21,27)(22,25)(24,32)(28,30)(29,31);(10,13)(14,16)(18,20)(19,25)(22,30)(23,31)(26,32)(27,28);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20,21,18)(19,26,27,23)(22,29,30,24)(25,31,28,32);(3,8)(5,7)(19,25)(22,30)(23,32)(24,29)(26,31)(27,28)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72308624, #72308985 |
| maximal normal subgroup of | #72308624, #72308985 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6603248);