Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500401 |
| 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 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22)(23,31)(24,27)(25,34)(26,30)(28,32)(29,35)(33,36);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,25,24,26,32,35,33)(22,27,29,31,30,36,34,28);(7,13)(8,15)(10,16)(11,18)(21,24,25,32,26,33,35,23)(22,28,29,27,30,31,34,36);(5,8,9,15)(6,11,12,18)(7,14,13,19)(10,17,16,20);(7,13)(10,16)(14,19)(17,20)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=g,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ehi,d^c=dhi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72311798, #72314649, #72314836, … |
| maximal normal subgroup of | #72311798, #72314649, #72314836, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6434186);