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 | |
7134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10)(11,16)(12,14)(13,15)(17,18)(19,22)(20,24)(21,23);(1,2)(3,6)(4,7)(5,8)(9,11,13,14)(10,12,15,16)(17,19,21,24)(18,22,23,20);(1,3)(2,5)(4,8)(6,7)(9,11,13,14)(10,12,15,16)(17,19)(18,20)(21,24)(22,23);(9,11,13,14)(10,12,15,16)(17,20)(18,19)(21,22)(23,24);(1,3)(2,6)(4,8)(5,7)(9,10)(11,12)(13,15)(14,16);(1,2)(3,5)(4,7)(6,8)(9,12)(10,14)(11,15)(13,16)Polycyclic presentation:
2^9|b^2=h,c^2=g,d^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72233520, #72233541, #72233606, … |
| maximal normal subgroup of | #72233520, #72233541, #72233606, … |
| maximal quotient of | #72261296, #72263595 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7560257);