Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437753 |
| 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 | #34609 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1709 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(18,22)(20,28)(23,29)(24,30)(25,32)(27,31);(9,11)(10,13)(12,15)(14,16)(17,18,20,22,21,24,28,30)(19,25,27,32,26,23,31,29);(2,5)(6,8)(11,15)(13,16)(17,19,21,26)(18,23,24,25)(20,27,28,31)(22,29,30,32);(11,15)(13,16)(18,24)(19,26)(22,30)(27,31);(1,3,4,7)(2,6,5,8)(18,24)(22,30)(23,25)(29,32)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72256843, #72256848, #72256966, … |
| maximal normal subgroup of | #72256843, #72256848, #72256966, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366276);