Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| 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×#459 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1812 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22)(23,27)(24,29)(25,28)(26,30)(31,35)(32,34)(33,36);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23)(22,27)(24,32)(25,33)(26,31)(28,36)(29,34)(30,35);(1,2,3,4)(22,28)(23,31)(24,26)(27,34)(32,33)(35,36);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(21,24,25,26)(22,29,28,30)(23,32,33,31)(27,34,36,35);(7,15)(10,13)(14,20)(18,19)(23,31)(24,26)(27,35)(29,30)(32,33)(34,36)Polycyclic presentation:
2^9|a^2=f,c^2=gi,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72283216, #72283719, #72285043, … |
| maximal normal subgroup of | #72283216, #72283719, #72285043, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591406);