Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393928 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14777 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1307 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,8,14,7,13)(3,15,17,5,18,24,11,9)(4,10,21,27,23,28,22,29)(16,30,31,25,32,26,19,20);(1,3,6,11,8,18,7,17)(2,9,12,24,14,5,13,15)(4,19,21,32,23,31,22,16)(10,26,27,25,28,30,29,20);(1,4,6,21,8,23,7,22)(2,10,12,27,14,28,13,29)(3,16,11,19,18,32,17,31)(5,20,15,26,9,25,24,30);(1,5)(2,11)(3,13)(4,20)(6,15)(7,24)(8,9)(10,19)(12,18)(14,17)(16,29)(21,26)(22,30)(23,25)(27,32)(28,31)Polycyclic presentation:
2^9|a^2=e,b^2=ef,c^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 306406);