Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196038 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, prime radical, cosocle | C25 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#95 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6)(7,11)(8,9)(10,12)(13,14)(15,21)(16,24,26,19)(17,20)(18,23)(22,28,25,27);(1,2,4,3)(6,9)(10,11)(13,15)(14,18)(16,25,26,22)(17,23)(19,27,24,28)(20,21);(1,2,4,3)(13,16)(14,19)(15,22)(17,26)(18,27)(20,24)(21,28)(23,25);(1,3,4,2)(5,7,8,12)(6,10,9,11)(14,20)(18,21)(19,24)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=fg,d^2=ei,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^c=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 277764);