Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| 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×#3345 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1933 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(13,14)(15,22)(16,25)(17,20)(18,21)(19,24)(23,27)(26,28);(6,10)(7,12)(8,11)(13,15)(14,19)(16,23)(17,26)(18,24)(20,27)(21,22)(25,28);(1,2,3,4)(5,6,11,12,9,7,8,10);(6,7)(10,12)(13,16,18,17)(14,20,21,25)(15,23,24,26)(19,27,22,28);(5,7,11,10,9,6,8,12)(13,17,18,16)(14,20,21,25)(15,23,24,26)(19,28,22,27)Polycyclic presentation:
2^9|c^2=ghi,d^2=f,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72256965, #72256972, #72257050 |
| maximal normal subgroup of | #72256965, #72256972, #72257050 |
| maximal quotient of | #72264564 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6723925);