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 | |
7161 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,5)(6,7)(9,10)(11,12)(13,16)(14,15)(17,18,21,23)(19,24,22,20);(1,2)(3,7)(4,5)(6,8)(10,15)(11,13)(17,19)(18,20)(21,22)(23,24);(1,2,4,5)(3,6,8,7)(9,11,14,13)(10,12,15,16)(17,20,21,24)(18,22,23,19);(1,2)(3,6)(4,5)(7,8)(9,12)(10,13)(11,15)(14,16)(17,20)(18,19)(21,24)(22,23);(1,2)(3,7)(4,5)(6,8)(9,12)(10,11)(13,15)(14,16)(17,19)(18,20)(21,22)(23,24);(1,3,4,8)(2,6,5,7)(9,13,14,11)(10,16,15,12)Polycyclic presentation:
2^9|a^2=gi,c^2=g,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72232278, #72232841, #72241225, … |
| maximal normal subgroup of | #72232278, #72232841, #72241225, … |
| maximal quotient of | #72232426, #72232589, #72232923, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7890400);