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 | |
5934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10)(11,14)(12,16)(13,15)(18,22)(23,24);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,15)(13,16)(17,18,21,22)(19,23,20,24);(2,6)(4,7)(10,15)(11,16)(17,19,21,20)(18,23,22,24);(1,3)(2,4)(5,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,20,21,19)(18,24,22,23);(1,3)(2,4)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16);(1,4)(2,3)(5,7)(6,8)(9,10)(11,12)(13,15)(14,16)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72279819, #72304324, #72304347, … |
| maximal normal subgroup of | #72279819, #72304324, #72304347, … |
| maximal quotient of | #72280697, #72280741, #72281088, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7711614);