Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | #34937 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1464 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)(5,6,10,11,9,7,8,12)(13,14)(15,21,25,28)(16,18)(17,20)(19,24,23,27)(22,26);(5,7,9,6)(8,12,10,11)(13,15,22,24,18,25,17,27)(14,19,26,21,16,23,20,28);(2,4)(5,8,9,10)(6,11,7,12)(13,16,18,14)(15,23,25,19)(17,26,22,20)(21,24,28,27);(14,16)(19,23)(20,26)(21,28)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^b=eh,g^b=gh,d^c=dhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 100726);