Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(9,10,12,13)(11,14,15,16)(17,18,21,24)(19,27,29,32)(20,26,22,25)(23,30,31,28);(1,2,3,4)(5,6)(7,8)(17,19,22,30,21,29,20,28)(18,23,26,32,24,31,25,27);(6,8)(10,13)(14,16)(18,24)(23,31)(25,26)(27,32);(2,4)(9,11,12,15)(10,14,13,16)(18,25)(19,28)(20,22)(23,31)(24,26)(29,30)Polycyclic presentation:
2^9|a^2=f,b^2=ei,d^2=fi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=dei,e^d=ei,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72311481, #72311530, #72311534, … |
| maximal normal subgroup of | #72311481, #72311530, #72311534, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335691);