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 | |
10684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2,3,4)(5,6,9,12)(7,11,10,8);(1,2)(3,4)(13,14)(15,18)(16,19)(17,20);(5,7)(6,8)(9,10)(11,12)(13,15)(14,17)(16,20)(18,19);(5,8)(6,10)(7,12)(9,11)(13,14)(15,18)(16,19)(17,20);(5,8)(6,7)(9,11)(10,12)(13,14)(15,17)(16,19)(18,20);(5,7)(6,11)(8,12)(9,10)(13,15)(14,18)(16,20)(17,19)Polycyclic presentation:
2^9|a^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,f^e=fhi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72214565, #72219286, #72225797, … |
| maximal normal subgroup of | #72214565, #72219286, #72225797, … |
| maximal quotient of | #72214271, #72219262, #72232091, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7566503);