Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2537 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(7,11)(10,12)(13,14)(15,22)(16,19)(17,21)(18,26)(20,24)(23,27)(25,28);(2,3)(6,9)(7,11);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,23)(17,24)(18,27)(19,25)(21,28)(22,26);(1,3,4,2)(6,9)(10,12)(13,16,19,14)(15,23,25,20)(17,21,18,26)(22,24,28,27);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,21,16,26)(15,24,25,27)(20,28,23,22)Polycyclic presentation:
2^9|a^2=f,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72261143, #72261189, #72261248, … |
| maximal normal subgroup of | #72261143, #72261189, #72261248, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7053134);