Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396094 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,10,14,17)(8,15)(9,12)(11,16)(13,19,20,18);(1,2)(3,4)(5,7)(6,10,12,17)(8,13)(9,14)(11,18,15,19)(16,20);(2,4)(10,17)(18,19);(5,8)(6,11,12,15)(7,13)(9,16)(10,18,17,19)(14,20)Polycyclic presentation:
2^9|a^2=f,b^2=ef,d^2=ef,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225620, #72225630, #72225634, … |
| maximal normal subgroup of | #72225620, #72225630, #72225634, … |
| maximal quotient of | #72251551, #72265795, #72265797, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308256);