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 | D4×#120 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7659 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,7)(3,4,6,8)(9,10,13,12)(11,15,16,14)(17,18)(19,22)(20,23)(21,24);(1,3)(2,4)(5,6)(7,8)(9,11,13,16)(10,14,12,15)(17,19,21,20)(18,22,24,23);(1,4)(2,6)(3,7)(5,8)(9,11)(10,15)(12,14)(13,16)(17,19)(18,23)(20,21)(22,24);(17,18)(19,22)(20,23)(21,24);(1,3)(2,4)(5,6)(7,8)(9,12,13,10)(11,15,16,14)(17,20,21,19)(18,23,24,22);(1,4)(2,6)(3,7)(5,8)(9,11)(10,15)(12,14)(13,16)(17,18)(19,23)(20,22)(21,24)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,e^2=gh,b^a=bg,c^a=ch,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi125 conjugacy classes — view the full character table.
| maximal subgroup of | #72241225, #72241231, #72241234, … |
| maximal normal subgroup of | #72241225, #72241231, #72241234, … |
| maximal quotient of | D4×#2358, #72335285, #72335337, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7606947);