Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394180 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1090 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,8,10)(7,9,11,12)(13,14)(15,19)(16,17)(18,20)(21,22)(23,29)(24,33)(25,27)(26,31)(28,30)(32,35)(34,36);(5,7)(6,9)(8,11)(10,12)(14,17)(18,19)(21,23,30,36,25,32,24,31)(22,26,33,35,27,34,28,29);(2,4)(7,11)(9,12)(21,22)(23,26)(24,28)(25,27)(29,31)(30,33)(32,34)(35,36);(13,15,16,20)(14,18,17,19)(22,27)(26,34)(28,33)(29,35)Polycyclic presentation:
2^9|a^2=g,b^2=eh,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264686, #72264692, #72264751, … |
| maximal normal subgroup of | #72264686, #72264692, #72264751, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324001);