Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| 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 | #23860 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1924 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)(4,8)(5,6)(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,29)(21,24)(22,25)(26,32)(27,31)(28,30);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,28)(21,26)(22,27)(24,32)(25,31)(29,30);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(18,24)(19,26)(25,29)(27,28);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(18,24)(19,27)(20,22)(23,30)(26,28)(31,32);(2,6)(7,8)(17,20,21,22)(18,25,24,29)(19,28,26,27)(23,31,32,30)Polycyclic presentation:
2^9|a^2=gi,c^2=f,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72308931, #72308983 |
| maximal normal subgroup of | #72308931, #72308983 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7433028);