Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14,16,22,18,21,17,19)(15,20,25,28,24,27,23,26);(2,4)(14,19)(15,23)(16,17)(20,27)(21,22)(24,25);(6,9)(10,12)(13,15,16,25,18,24,17,23)(14,20,22,28,21,27,19,26);(13,15,17,23,18,24,16,25)(14,20,19,26,21,27,22,28);(5,7,8,11)(6,10,9,12)(14,21)(15,24)(19,22)(23,25)Polycyclic presentation:
2^9|a^2=f,c^2=hi,d^2=h,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72250494, #72256919, #72257055, … |
| maximal normal subgroup of | #72250494, #72256919, #72257055, … |
| maximal quotient of | #72264404, #72264691 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374764);