Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2818 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2615 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,8,9)(7,11,12,10)(13,14)(15,19)(16,17)(18,20)(21,22)(23,27)(24,28)(25,26)(29,30)(31,35)(32,36)(33,34);(6,9)(7,12)(13,14,16,17)(15,19,20,18)(22,26)(23,29)(28,34)(31,32);(6,9)(10,11)(14,17)(18,19)(21,23,25,29)(22,27,26,30)(24,32,33,31)(28,36,34,35);(1,2)(3,4)(5,7,8,12)(6,10,9,11)(13,15,16,20)(14,18,17,19)(21,22)(23,30)(24,28)(25,26)(27,29)(31,36)(32,35)(33,34);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,24,25,33)(22,28,26,34)(23,31,29,32)(27,35,30,36)Polycyclic presentation:
2^9|a^2=fh,b^2=fgh,c^2=fg,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5120688);