#6625599

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573715
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂×H(5,𝔽₂)	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

2929 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=h,e^2=hi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,e^d=eh

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, 6532795);