#7502547

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500740
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	#30432	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1726 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)(9,10,16,21)(11,17,24,12)(13,28,29,36)(14,31,15,20)(18,27,23,26)(19,32,35,30)(22,37,38,40)(25,34,33,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,17)(12,21)(13,22)(14,23)(15,18)(16,24)(19,33)(20,27)(25,35)(26,31)(28,37)(29,38)(30,34)(32,39)(36,40);(9,12,16,17)(10,18,21,23)(11,20,24,31)(13,25,29,33)(14,27,15,26)(19,34,35,39)(22,36,38,28)(30,37,32,40);(2,7)(3,8)(9,13)(10,19)(11,22)(12,25)(14,32)(15,30)(16,29)(17,33)(18,34)(20,36)(21,35)(23,39)(24,38)(26,37)(27,40)(28,31);(1,4,5,6)(2,3,7,8)(13,29)(19,35)(22,38)(25,33)(28,36)(30,32)(34,39)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7409743);