Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2173 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,8)(9,11)(10,12)(13,14,17,20,16,21,25,19)(15,22,24,27,23,18,26,28);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,16,23)(14,18,21,22)(17,24,25,26)(19,27,20,28);(2,4)(14,19)(17,25)(18,27)(20,21)(22,28)(24,26);(6,11)(7,12)(14,20)(15,23)(17,25)(18,27)(19,21)(22,28);(5,8,9,10)(6,7,11,12)(14,21)(15,23)(19,20)(24,26)Polycyclic presentation:
2^9|a^2=g,b^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72310805, #72310815, #72310835, … |
| maximal normal subgroup of | #72310805, #72310815, #72310835, … |
| maximal quotient of | #72311486, #72311831, #72311837 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411302);