Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
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 | |
6252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,4)(3,7,8,6)(9,10,13,15)(11,16,12,14);(1,3)(2,6)(4,7)(5,8)(11,12)(14,16)(17,18)(19,20)(21,23)(22,24);(9,11,13,12)(10,14,15,16)(17,19)(18,22)(20,23)(21,24);(2,4)(6,7)(11,12)(14,16)(17,19)(18,20)(21,24)(22,23);(1,4,5,2)(3,7,8,6)(9,12,13,11)(10,14,15,16)(17,20,21,22)(18,19,23,24);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,19)(18,20)(21,24)(22,23)Polycyclic presentation:
2^9|a^2=gh,c^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232290, #72256180, #72262986, … |
| maximal normal subgroup of | #72232290, #72256180, #72262986, … |
| maximal quotient of | #72232380, #72232389, #72323352, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10051222);