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 |
| prime radical | #36258 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5324 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)(11,15)(12,16)(13,14)(17,18,21,20)(19,23,24,22);(1,2)(3,6)(4,7)(5,8)(10,14)(11,16)(17,19,21,24)(18,22,20,23);(1,3)(2,5)(4,8)(6,7)(9,11)(10,12)(13,16)(14,15)(17,18,21,20)(19,22,24,23);(1,2)(3,6)(4,7)(5,8)(9,12)(10,11)(13,15)(14,16)(17,20,21,18)(19,23,24,22);(1,2)(3,5)(4,7)(6,8);(1,3)(2,6)(4,8)(5,7)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,f^e=fhi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72238571, #72238590, C2×#7660914 |
| maximal normal subgroup of | #72238571, #72238590, C2×#7660914 |
| maximal quotient of | C2×#7660914 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7568110);