Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #36101 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6130 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)(3,7)(4,8)(5,6)(9,10,13,16)(11,14,12,15)(17,18)(19,23)(20,24)(21,22);(2,6)(7,8)(9,11,13,12)(10,14,16,15)(18,22)(19,20);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,16,14)(17,19,21,20)(18,23,22,24);(9,12,13,11)(10,15,16,14)(19,20)(23,24);(1,3,5,4)(2,7,6,8)(17,20,21,19)(18,24,22,23);(1,4,5,3)(2,8,6,7)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,c^b=ci,e^b=ei,d^c=di,e^d=ei140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537635);