#2618259

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
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

1739 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=fgh,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,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, 2525455);