Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| 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×#3345 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(13,14,18,19,17,22,16,21)(15,23,26,28,24,20,25,27);(5,6,8,9)(7,10,11,12)(14,19)(16,18)(20,27)(21,22)(23,28)(25,26);(1,2,3,4)(6,9)(7,11)(13,15)(14,20)(16,25)(17,24)(18,26)(19,27)(21,28)(22,23);(5,7)(6,10)(8,11)(9,12)(15,24)(20,23)(25,26)(27,28);(6,9)(7,11)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=gi,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh83 conjugacy classes — view the full character table.
| maximal subgroup of | #72257285, #72257294, #72257414, … |
| maximal normal subgroup of | #72257285, #72257294, #72257414, … |
| maximal quotient of | #72264316, #72307880 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6513900);