Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460859 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16373 |
| 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 | |
1046 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)(3,8,15,25)(4,10)(5,19)(6,12)(7,21)(9,18)(11,13)(14,29,17,23)(16,31,28,24)(20,27)(22,32,26,30);(1,3,13,28,6,15,7,16)(2,8,11,24,12,25,21,31)(4,17,20,30,5,14,18,32)(9,26,10,23,27,22,19,29);(1,4)(2,9)(3,14)(5,6)(7,20)(8,22)(10,21)(11,19)(12,27)(13,18)(15,17)(16,32)(23,24)(25,26)(28,30)(29,31);(1,5,6,4)(2,10)(3,14,15,17)(7,20,13,18)(8,23)(9,21)(11,27)(12,19)(16,30,28,32)(22,24)(25,29)(26,31)Polycyclic presentation:
2^9|a^2=g,b^2=ef,d^2=e,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,f^c=fi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368055);