Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459338 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16338 |
| 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 | |
1148 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(3,12)(4,17)(5,6)(9,24)(10,23)(13,22)(14,28)(15,16)(18,30)(25,29)(31,32);(1,2,5,21,7,8,6,11)(3,13,12,9,16,24,15,22)(4,10,20,26,17,23,19,27)(14,29,28,25,18,31,30,32);(1,3)(2,9)(4,18)(5,15)(6,12)(7,16)(8,22)(10,25)(11,24)(13,21)(14,17)(19,30)(20,28)(23,32)(26,29)(27,31);(1,4,5,20,7,17,6,19)(2,10,21,26,8,23,11,27)(3,14,12,28,16,18,15,30)(9,25,24,31,22,32,13,29)Polycyclic presentation:
2^9|b^2=ef,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72319982, #72320000, #72320106, … |
| maximal normal subgroup of | #72319982, #72320000, #72320106, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 366534);