Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393944 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,17)(18,20)(21,22)(23,27)(24,33)(25,28)(26,29)(30,35)(31,34)(32,36);(6,9)(10,12)(21,23,25,32)(22,27,28,36)(24,31,26,30)(29,35,33,34);(2,4)(5,7,8,11)(6,10,9,12)(14,17)(18,19)(22,28)(23,30)(24,26)(27,34)(31,32)(35,36);(13,15,16,20)(14,18,17,19)(21,24,25,26)(22,29,28,33)(23,31,32,30)(27,35,36,34)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=e,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,d^c=di,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264490, #72264500, #72264567, … |
| maximal normal subgroup of | #72264490, #72264500, #72264567, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 322282);