Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469989 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
2050 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,24)(19,23,26,30)(20,22,28,25)(27,29,32,31);(9,11)(10,13)(12,15)(14,16)(18,22)(20,28)(23,29)(24,25)(27,32)(30,31);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(17,19,21,26)(18,23,24,30)(20,27,28,32)(22,29,25,31);(11,15)(13,16)(18,24)(19,26)(22,25)(27,32);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(19,26)(23,30)(27,32)(29,31)Polycyclic presentation:
2^9|a^2=i,c^2=gi,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285539 |
| maximal normal subgroup of | #72285539 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6383928);