Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393928 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14811 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1674 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)(3,5,17,23)(4,10)(6,12)(7,13)(8,14)(9,16,22,11)(15,26,31,19)(18,25,27,32)(20,28)(21,30)(24,29);(1,3,6,11,8,17,7,16)(2,9,12,23,14,22,13,5)(4,18,20,15,21,27,24,31)(10,26,28,32,30,19,29,25);(1,4)(2,10)(3,15)(5,19)(6,20)(7,24)(8,21)(9,25)(11,27)(12,28)(13,29)(14,30)(16,18)(17,31)(22,32)(23,26);(1,5,6,9,8,23,7,22)(2,11,12,17,14,16,13,3)(4,19,20,25,21,26,24,32)(10,27,28,31,30,18,29,15)Polycyclic presentation:
2^9|a^2=f,b^2=ef,d^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.
| maximal subgroup of | #72318824, #72318894, #72318940, … |
| maximal normal subgroup of | #72318824, #72318894, #72318940, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308269);