Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437608 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23430 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2287 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,7)(3,8,4,6)(9,10,14,19)(11,20,17,16)(12,21,24,22)(13,23,15,18);(1,3,5,4)(2,6,7,8)(9,11,14,17)(10,16,19,20)(12,23,24,18)(13,22,15,21);(9,12)(10,17)(11,19)(13,16)(14,24)(15,20)(18,22)(21,23);(1,4,5,3)(2,6,7,8)(9,10,14,19)(11,12,17,24)(13,18,15,23)(16,22,20,21);(9,13,14,15)(10,18,19,23)(11,21,17,22)(12,16,24,20)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213480, #72213818 |
| maximal normal subgroup of | #72213480, #72213818 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346937);