Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2090 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(6,7)(9,10,11,12)(15,23)(16,17)(19,27)(21,22)(24,28)(25,26);(5,6,8,7)(13,14)(15,24)(16,22)(17,21)(18,20)(19,26)(23,28)(25,27);(2,4)(5,7,8,6)(13,15)(14,19)(16,26)(17,23)(18,25)(20,28)(21,24)(22,27);(14,20)(15,25)(21,22)(23,26)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,d^b=di,f^b=fi,g^b=gi,d^c=di,f^c=fi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72214149, #72215962, #72216022 |
| maximal normal subgroup of | #72214149, #72215962, #72216022 |
| maximal quotient of | #72214111 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265288);