Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667652 |
| 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 | C2×#2535 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2148 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,24,20,25,21,26)(19,27,30,32,28,23,29,31);(2,5)(3,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,27)(26,32);(11,16)(13,15)(18,24)(21,22)(23,31)(25,26)(27,32)(29,30);(2,5)(6,8)(10,14)(13,15)(18,25)(23,27)(24,26)(31,32);(1,3,4,7)(2,6,5,8)(19,28)(23,27)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=gi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,g^c=gi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72256895, #72256920, #72256993 |
| maximal normal subgroup of | #72256895, #72256920, #72256993 |
| maximal quotient of | #72308932 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6702049);