Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #463531 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16476 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
944 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,8,6,12,5,11)(3,13,20,22,14,9,16,28)(4,10,18,26,19,27,21,23)(15,29,17,31,30,24,32,25);(2,8)(3,14)(5,7)(9,22)(10,23)(11,12)(13,28)(15,30)(18,21)(24,25)(26,27)(29,31);(1,3,7,16,6,14,5,20)(2,9,11,28,12,13,8,22)(4,17,21,30,19,32,18,15)(10,25,26,24,27,31,23,29);(1,4,6,19)(2,10)(3,15)(5,18,7,21)(8,23)(9,24,13,29)(11,26)(12,27)(14,30)(16,17)(20,32)(22,25,28,31)Polycyclic presentation:
2^9|a^2=efg,c^2=ef,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,d^c=de,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251676, #72251689, #72251697, … |
| maximal normal subgroup of | #72251676, #72251689, #72251697, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 370727);