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 | #42475 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,6,5)(3,4,8,7)(10,13)(14,16)(17,18)(19,20)(21,22)(23,24);(1,3)(2,7)(4,5)(6,8)(9,10)(11,14)(12,13)(15,16);(2,5)(3,8)(18,22)(19,23);(1,2,6,5)(3,7,8,4)(17,19,21,23)(18,20,22,24);(1,4)(2,3)(5,8)(6,7)(17,20)(18,23)(19,22)(21,24);(1,5,6,2)(3,4,8,7)(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=i,d^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72303495, #72303497 |
| maximal normal subgroup of | #72303495, #72303497 |
| maximal quotient of | #72303442, #72303524 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7652549);