Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #428059 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2002 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1341 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(6,11)(7,15)(8,9)(12,14)(13,19)(18,20);(1,2)(3,4)(6,12,14,11)(13,18,20,19);(2,4)(5,6)(7,13)(8,12)(9,11)(10,14)(15,20)(16,19)(17,18);(5,7,8,16,10,15,9,17)(6,13,12,19,14,20,11,18)Polycyclic presentation:
2^9|b^2=e,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72287326, #72287348, #72287360, … |
| maximal normal subgroup of | #72287326, #72287348, #72287360, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335255);