Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2)(3,4)(8,12)(10,11)(14,18)(16,19);(5,6,9,7)(8,10,12,11)(13,14)(15,19)(16,20)(17,18);(2,4)(6,7)(10,11)(14,18)(15,20);(1,2)(3,4);(5,7,9,6)(8,11,12,10)(13,15,17,20)(14,19,18,16);(5,8)(6,10)(7,11)(9,12)(13,16,17,19)(14,20,18,15)Polycyclic presentation:
2^9|b^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^c=dh,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72305362, #72305378, #72311840 |
| maximal normal subgroup of | #72305362, #72305378, #72311840 |
| maximal quotient of | #72306203, #72306208 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7579075);