Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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×#3267 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2069 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6,9,10)(7,11,8,12)(13,14,17,18)(15,19,23,20)(16,21,25,26)(22,27,28,24);(2,3)(6,10)(11,12)(13,15,17,23)(14,19,18,20)(16,24,25,27)(21,22,26,28);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,15)(14,20)(16,22)(17,23)(18,19)(21,24)(25,28)(26,27);(1,3,4,2)(5,7,9,8)(6,11,10,12)(13,16)(14,21)(15,22)(17,25)(18,26)(19,27)(20,24)(23,28);(5,8,9,7)(6,11,10,12)(15,23)(19,20)(22,28)(24,27)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fgi,d^2=fgi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72263913, #72263935, #72263980 |
| maximal normal subgroup of | #72263913, #72263935, #72263980 |
| maximal quotient of | #72264198 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7473236);