Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994107 |
| 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 | #26584 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1975 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6,9,11,10,7,8,12)(16,24)(19,25)(23,28)(26,27);(2,3)(13,14,17,20)(15,21,22,18)(16,25,24,19)(23,26,28,27);(5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,23)(17,22)(19,26)(20,21)(24,28)(25,27);(1,3,4,2)(5,8,10,9)(6,11,7,12)(15,22)(18,21)(23,28)(26,27);(5,8,10,9)(6,11,7,12)(13,16,17,24)(14,19,20,25)(15,23,22,28)(18,26,21,27)Polycyclic presentation:
2^9|a^2=fg,b^2=h,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72283216, #72283258, #72283523, … |
| maximal normal subgroup of | #72283216, #72283258, #72283523, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7090912);