Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #115010 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1315 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,8,6,4)(5,7)(9,10)(11,18)(12,23)(13,16)(14,17)(15,20)(19,22)(21,24);(1,3,5,6)(2,4,7,8)(9,11,14,22)(10,15,17,24)(12,21,13,20)(16,19,23,18);(1,4)(2,6)(3,7)(5,8)(9,10)(11,19)(12,16)(13,23)(14,17)(15,20)(18,22)(21,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,d^c=dhi,f^c=fi,g^c=ghi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72233811, #72233814, #72233815 |
| maximal normal subgroup of | #72233811, #72233814, #72233815 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22207);