#7545929

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519063
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	#25434	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1708 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 7453125);