Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7644 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(2,3)(5,6)(7,11)(8,12)(9,10)(14,17)(18,20);(1,2,4,3)(6,10)(11,12);(2,3)(13,14)(15,18)(16,17)(19,20);(5,7,9,8)(6,11,10,12)(14,17)(18,20);(1,3,4,2)(5,8,9,7)(6,12,10,11);(13,15,16,19)(14,18,17,20)Polycyclic presentation:
2^9|b^2=i,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^c=dh,e^c=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72238211, #72238212, #72238660, … |
| maximal normal subgroup of | #72238211, #72238212, #72238660, … |
| maximal quotient of | #72265611, #72303401, #72316559 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7574147);