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 | C2×#3449 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,19)(16,23)(17,18)(20,24)(21,25)(22,28)(26,27);(6,9)(10,11)(15,21)(19,25)(22,27)(26,28);(5,6)(7,10)(8,9)(11,12)(14,18)(19,25)(20,23)(26,28);(5,7)(6,10)(8,12)(9,11)(13,14)(15,19)(16,20)(17,18)(21,25)(22,26)(23,24)(27,28);(13,15)(14,19)(16,22)(17,21)(18,25)(20,26)(23,28)(24,27);(13,16,17,24)(14,20,18,23)(15,22,21,27)(19,26,25,28)Polycyclic presentation:
2^9|a^2=i,f^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 | #72256630, #72256632, #72256654, … |
| maximal normal subgroup of | #72256630, #72256632, #72256654, … |
| maximal quotient of | #72263742, #72263747 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533574);