#7597034

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563319
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61086	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1960 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)(3,7)(4,5)(6,8)(9,10,16,22)(11,23,28,34)(12,19,31,37)(13,18,30,38)(14,21,15,20)(17,27,35,26)(24,33,25,32)(29,36,40,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,26,16,28,15,27)(10,17,20,23,22,35,21,34)(12,29,30,25,31,40,13,24)(18,36,37,33,38,39,19,32);(2,5)(6,7)(9,12,14,30,16,31,15,13)(10,18,20,37,22,38,21,19)(11,24,27,40,28,25,26,29)(17,32,34,39,35,33,23,36);(1,3,4,8)(2,6,5,7)(9,12,14,30,16,31,15,13)(10,18,20,37,22,38,21,19)(11,25,27,29,28,24,26,40)(17,33,34,36,35,32,23,39);(9,13,15,31,16,30,14,12)(10,19,21,38,22,37,20,18)(11,24,27,40,28,25,26,29)(17,32,34,39,35,33,23,36)

Polycyclic presentation:

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

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, 7504230);