Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459299 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2752 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16707 |
| prime radical | C25 | #41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11,15,7)(4,16)(5,20)(6,9)(8,18)(10,12)(13,28,17,31)(14,22,26,25)(19,30)(21,29,23,32)(24,27);(1,3,12,26,6,15,5,14)(2,7,10,22,9,11,20,25)(4,17,18,32,19,13,27,29)(8,23,30,31,24,21,16,28);(1,4,5,18,6,19,12,27)(2,8,10,16,9,24,20,30)(3,13,14,29,15,17,26,32)(7,21,22,31,11,23,25,28);(2,9)(7,11)(10,20)(22,25)Polycyclic presentation:
2^9|a^2=g,b^2=efh,c^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,d^c=de,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72251914, #72251917, #72251922, … |
| maximal normal subgroup of | #72251914, #72251917, #72251922, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378082);