Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455577 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16266 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23439 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1127 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)(4,17)(5,9)(6,11)(7,12)(8,16)(10,19)(14,22)(15,23)(18,26)(20,25)(21,24)(27,30)(28,32)(29,31);(1,3,5,16,7,14,6,21)(2,8,11,13,12,24,9,22)(4,15,19,29,20,28,18,30)(10,23,17,27,26,32,25,31);(2,9)(3,14)(4,18)(5,6)(8,22)(10,26)(11,12)(13,24)(15,29)(19,20)(23,32)(28,30);(1,4)(2,10)(3,15)(5,19)(6,18)(7,20)(8,23)(9,25)(11,17)(12,26)(13,27)(14,28)(16,29)(21,30)(22,31)(24,32)Polycyclic presentation:
2^9|b^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362773);