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 | |
6670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,6)(3,8)(9,10,13,14)(11,15,16,12)(17,18,21,23)(19,20,24,22);(1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,16)(14,15);(1,3)(2,7)(4,6)(5,8)(10,14)(12,15)(17,19,21,24)(18,22,23,20);(1,2)(3,4)(5,6)(7,8)(9,12)(10,11)(13,15)(14,16)(17,19)(18,20)(21,24)(22,23);(1,2)(3,7)(4,8)(5,6)(9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,22)(23,24);(1,4,5,7)(2,3,6,8)Polycyclic presentation:
2^9|a^2=hi,c^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72335453 |
| maximal normal subgroup of | #72335453 |
| maximal quotient of | #72335484, #72335731, #72335736 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7825889);