Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806527 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,17,19)(15,23,25,20)(16,21,26,27)(18,22,24,28);(5,7)(6,9)(8,12)(10,11)(14,19)(15,16)(18,24)(20,21)(23,27)(25,26);(1,2,3,4)(13,15,24,26,17,25,18,16)(14,20,28,21,19,23,22,27);(6,10)(9,11)(14,19)(20,23)(21,27)(22,28);(7,12)(9,11)(13,16,18,25,17,26,24,15)(14,21,22,20,19,27,28,23)Polycyclic presentation:
2^9|a^2=i,c^2=gh,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6721564);