Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4462490 |
| automorphism | Aut(G) order 33554432 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2118 conjugacy classes of subgroups; only the 554 normal subgroups are shown.
This group has 2118 conjugacy classes of subgroups — too many to draw. Only its 554 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(6,9)(7,11)(14,16)(15,19)(21,22,26,30)(23,28,24,27)(25,33,34,29)(31,35,32,36);(7,11)(10,12)(13,14,17,16)(15,18,19,20)(21,23,26,24)(22,27,30,28)(25,31,34,32)(29,35,33,36);(5,6,8,9)(7,10,11,12)(13,15)(14,18)(16,20)(17,19)(21,22)(23,28)(24,27)(25,29)(26,30)(31,36)(32,35)(33,34);(1,2,3,4)(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,18,19,20)(21,24,26,23)(22,28,30,27)(25,32,34,31)(29,36,33,35);(21,25,26,34)(22,29,30,33)(23,31,24,32)(27,35,28,36)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fg,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfg,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4371353);