Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62862 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C22×#535 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
9250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14,18,15)(16,19,23,21)(17,22,25,20)(24,27,28,26);(6,9)(10,11)(16,23)(19,21)(24,28)(26,27);(5,6)(7,10)(8,9)(11,12)(14,15)(19,21)(20,22)(26,27);(5,7)(6,10)(8,12)(9,11)(13,15,18,14)(16,21,23,19)(17,22,25,20)(24,27,28,26);(13,16)(14,19)(15,21)(17,24)(18,23)(20,26)(22,27)(25,28);(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)Polycyclic presentation:
2^9|a^2=i,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh,d^c=dh140 conjugacy classes — view the full character table.
| maximal subgroup of | #72256610, #72256611, #72256612, … |
| maximal normal subgroup of | #72256610, #72256611, #72256612, … |
| maximal quotient of | #72263415, #72263548, #72323336, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533551);