Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454389 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2815 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16660 |
| prime radical | #23443 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1178 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)(3,7)(4,14)(5,18)(6,9)(8,17)(10,20)(11,27)(12,21)(13,26)(15,30)(16,29)(19,23)(22,28)(24,31)(25,32);(7,21)(22,24)(23,26)(27,30);(1,3)(2,7)(4,15,16,11)(5,19)(6,12)(8,24,25,22)(9,21)(10,26)(13,20)(14,27,29,30)(17,28,32,31)(18,23);(1,4)(2,8,9,25)(3,11)(5,17)(6,16)(7,22,21,24)(10,14,18,29)(12,15)(13,28)(19,31)(20,32)(23,27,26,30)Polycyclic presentation:
2^9|c^2=f,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72237054, #72237084, #72237661, … |
| maximal normal subgroup of | #72237054, #72237084, #72237661, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376876);