#4171523

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1958 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2,5,3)(4,7,8,6)(9,10,13,14)(11,12,16,15)(17,18)(19,23)(20,22)(21,24)(25,26,29,30)(27,33,35,31)(28,36,38,32)(34,39,37,40);(4,8)(6,7)(9,11,13,16)(10,12,14,15)(17,18)(19,23)(20,22)(21,24)(26,30)(31,33)(32,36)(39,40);(2,3)(6,7)(10,14)(12,15)(17,18,20,22)(19,21,24,23)(25,27,29,35)(26,31,30,33)(28,37,38,34)(32,40,36,39);(1,3,5,2)(4,7,8,6)(9,12,13,15)(10,11,14,16)(17,18)(19,21)(20,22)(23,24)(25,28,29,38)(26,32,30,36)(27,34,35,37)(31,39,33,40);(1,4)(2,6)(3,7)(5,8)(9,11)(10,15)(12,14)(13,16)(17,19)(18,21)(20,24)(22,23)(25,26)(27,33)(28,36)(29,30)(31,35)(32,38)(34,39)(37,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=fi,c^2=fh,d^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4078719);