#7503260

order	512
nilpotency class	3
isoclinism	isoclinic class of #7501009
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60366	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2637 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)(4,6)(5,8)(9,10,14,32,15,20,16,19)(11,21,23,30,18,13,24,31)(12,25,33,17,29,36,28,37)(22,34,26,38,35,40,39,27);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,26)(14,23)(15,18)(16,24)(17,34)(19,30)(20,21)(22,28)(25,27)(29,39)(31,32)(33,35)(36,38)(37,40);(9,12)(10,17)(11,22)(13,27)(14,28)(15,29)(16,33)(18,35)(19,36)(20,37)(21,38)(23,39)(24,26)(25,32)(30,34)(31,40);(2,6)(3,8)(9,13)(10,18)(11,20)(12,27)(14,30)(15,21)(16,31)(17,35)(19,24)(22,37)(23,32)(25,39)(26,36)(28,34)(29,38)(33,40);(9,14,15,16)(10,19,20,32)(11,23,18,24)(12,28,29,33)(13,30,21,31)(17,36,37,25)(22,39,35,26)(27,34,38,40)

Polycyclic presentation:

2^9|a^2=gi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,g^c=gi,h^c=hi,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, 7410456);