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 | |
5788 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)(3,4)(5,7)(6,8)(9,10,13,14)(11,15,16,12)(17,18,21,19)(20,23,24,22);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,16)(14,15)(18,19)(22,23);(9,10)(11,12)(13,14)(15,16)(17,19,21,18)(20,23,24,22);(2,7)(4,6)(10,14)(12,15)(17,20)(18,22)(19,23)(21,24);(1,4,5,6)(2,3,7,8)(9,12,13,15)(10,11,14,16);(1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,16)(14,15)Polycyclic presentation:
2^9|a^2=hi,c^2=i,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 | #72327258, #72328499, #72336994 |
| maximal normal subgroup of | #72327258, #72328499, #72336994 |
| maximal quotient of | #72327316, #72327712, #72328159, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10065605);