Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #469778 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16662 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1067 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)(3,13)(4,18)(5,11)(6,12)(7,10)(8,15)(9,19)(14,28)(16,24)(17,23)(20,26)(21,25)(22,30)(27,29)(31,32);(1,3,5,15,6,16,7,17)(2,8,10,13,12,23,11,24)(4,14,19,29,20,30,21,31)(9,22,18,27,25,28,26,32);(1,2)(3,8,16,23)(4,18,20,26)(5,10)(6,12)(7,11)(9,19,25,21)(13,17,24,15)(14,27)(22,31)(28,29)(30,32);(1,4,7,21,6,20,5,19)(2,9,10,18,12,25,11,26)(3,14,17,31,16,30,15,29)(8,22,13,27,23,28,24,32)Polycyclic presentation:
2^9|b^2=e,c^2=f,d^2=eg,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=de,g^c=gi,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251686, #72251691, #72251699, … |
| maximal normal subgroup of | #72251686, #72251691, #72251699, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376974);