#7503945

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂×D₄²
prime radical	#31071	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1764 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,g^c=gi,h^c=hi,g^d=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7411141);