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 | #39358 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5282 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10,14,13)(11,15,16,12)(17,18,21,24)(19,22,20,23);(1,2)(3,6)(4,5)(7,8)(9,11,14,16)(10,15,13,12)(17,18)(19,22)(20,23)(21,24);(2,4)(6,8)(10,13)(11,16)(17,19,21,20)(18,22,24,23);(3,7)(6,8)(9,10,14,13)(11,15,16,12)(17,20,21,19)(18,23,24,22);(1,3,5,7)(2,6,4,8)(9,12,14,15)(10,16,13,11);(1,4,5,2)(3,8,7,6)(9,13,14,10)(11,15,16,12)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=g,d^2=gh,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72303495, #72303497 |
| maximal normal subgroup of | #72303495, #72303497 |
| maximal quotient of | #72303399, #72303524 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7753832);