Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #400512 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2491 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14802 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,11,5,12)(3,13,16,25,17,24,15,8)(4,18,21,27,22,26,20,9)(14,23,30,31,19,32,29,28);(1,3,5,15,7,17,6,16)(2,8,10,13,11,25,12,24)(4,19,20,30,22,14,21,29)(9,23,18,31,27,32,26,28);(1,4)(2,9)(3,14)(5,21)(6,20)(7,22)(8,23)(10,26)(11,27)(12,18)(13,28)(15,30)(16,29)(17,19)(24,31)(25,32);(2,10,11,12)(8,13,25,24)(9,26,27,18)(23,28,32,31)Polycyclic presentation:
2^9|a^2=f,b^2=e,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319592, #72319976, #72320106, … |
| maximal normal subgroup of | #72319592, #72319976, #72320106, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 307708);