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 | |
7551 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)(3,5)(4,6)(7,8)(9,10,13,16)(11,12,14,15)(17,18)(19,22)(20,24)(21,23);(1,2,5,3)(4,7,8,6)(9,11)(10,12)(13,14)(15,16)(17,19,21,24)(18,20,23,22);(2,3)(6,7)(9,12)(10,14)(11,16)(13,15)(17,19)(18,22)(20,23)(21,24);(1,2,5,3)(4,6,8,7)(9,11)(10,15)(12,16)(13,14)(17,20,21,22)(18,19,23,24);(1,3,5,2)(4,6,8,7)(9,11)(10,12)(13,14)(15,16)(17,18)(19,20)(21,23)(22,24);(1,4)(2,6)(3,7)(5,8)(9,12)(10,14)(11,16)(13,15)(17,19)(18,22)(20,23)(21,24)Polycyclic presentation:
2^9|a^2=gi,b^2=g,d^2=g,e^2=h,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 | #72233590, #72233594, #72233606, … |
| maximal normal subgroup of | #72233590, #72233594, #72233606, … |
| maximal quotient of | #72262192 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7875371);