Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2228 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)(11,15)(12,14)(13,16)(17,18)(19,21)(20,24)(22,27)(23,25)(26,30)(28,31)(29,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,16)(14,15);(1,2)(3,5)(4,6)(7,8)(11,16)(13,15)(17,19,21,18,23,27,22,25)(20,26,30,24,29,31,28,32);(2,6)(5,7)(10,14)(13,15)(17,20,23,29)(18,24,25,32)(19,26,27,31)(21,30,22,28);(3,8)(5,7)(17,21,23,22)(18,19,25,27)(20,28,29,30)(24,31,32,26)Polycyclic presentation:
2^9|c^2=gi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72282013, #72282556, #72282746, … |
| maximal normal subgroup of | #72282013, #72282556, #72282746, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6815260);