Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×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 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1106 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)(7,10)(8,9)(11,12)(13,14)(15,18)(16,20)(17,19)(21,22)(23,28)(24,33)(25,29)(26,30)(27,31)(32,35)(34,36);(6,9)(10,12)(13,15,17,16)(14,18,19,20)(21,23,25,34,26,24,27,32)(22,28,29,36,30,33,31,35);(5,7)(6,10)(8,11)(9,12)(14,19)(18,20)(21,24,26,23)(22,28,30,33)(25,34,27,32)(29,35,31,36);(2,4)(7,11)(10,12)(13,16,17,15)(14,20,19,18)(21,25,26,27)(22,29,30,31)(23,32,24,34)(28,35,33,36)Polycyclic presentation:
2^9|b^2=fg,c^2=i,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72264950, #72265018, #72265097, … |
| maximal normal subgroup of | #72264950, #72265018, #72265097, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 373158);