Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418488 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1857 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(6,12)(8,14)(9,10)(13,15)(16,17)(19,20);(1,2)(3,4)(5,6)(7,15)(8,9)(10,12)(11,14)(13,17)(16,20)(18,19);(2,4)(5,7)(6,13)(8,15)(9,16)(10,17)(11,18)(12,19)(14,20);(5,8,10,14,11,12,9,6)(7,15,17,20,18,19,16,13)Polycyclic presentation:
2^9|d^2=ei,e^2=i,g^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,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72255815, #72256179, #72256189, … |
| maximal normal subgroup of | #72255815, #72256179, #72256189, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325685);