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 | |
6034 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,4)(3,7,8,6)(9,10,14,16)(11,13,15,12)(17,18,21,24)(19,22,20,23);(2,4)(6,7)(9,11)(10,12)(13,16)(14,15)(17,18)(19,22)(20,23)(21,24);(1,2,5,4)(3,6,8,7)(9,11,14,15)(10,12,16,13)(17,19,21,20)(18,22,24,23);(1,3)(2,6)(4,7)(5,8)(9,12,14,13)(10,15,16,11);(1,3)(2,7)(4,6)(5,8)(9,11)(10,12)(13,16)(14,15)(17,20,21,19)(18,23,24,22);(1,4,5,2)(3,7,8,6)(9,13,14,12)(10,15,16,11)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=h,e^2=gi,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 | #72300460, #72311493, #72312311, … |
| maximal normal subgroup of | #72300460, #72311493, #72312311, … |
| maximal quotient of | #72300521 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7895178);