Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #112057 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | #37618 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1095 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,7,4,3,8,6)(9,10)(11,17,19,23)(12,15)(13,16)(14,18,22,20)(21,24);(5,8)(6,7)(9,11)(10,14)(12,20)(13,19)(15,23)(16,22)(17,24)(18,21);(1,3)(2,4)(5,6)(7,8)(12,21)(14,22)(15,24)(18,20)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dhi,e^c=eh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72286774, #72287102, #72287107, … |
| maximal normal subgroup of | #72286774, #72287102, #72287107, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 19257);