Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2289 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,18,16)(15,19,23,25)(17,20,27,26)(21,22,28,24);(2,3)(5,7,8,11)(6,9,10,12)(13,15)(14,19)(16,25)(17,22)(18,23)(20,28)(21,26)(24,27);(1,2,4,3)(6,10)(9,12)(13,16,18,14)(15,21,23,28)(17,20,27,26)(19,22,25,24);(1,3,4,2)(15,22)(17,27)(19,28)(20,26)(21,25)(23,24);(15,23)(19,25)(21,28)(22,24)Polycyclic presentation:
2^9|a^2=fi,b^2=g,c^2=fi,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6348831);