#7501837

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500740
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	#60645	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2292 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)(3,5)(4,6)(7,8)(9,10,15,19)(11,21,23,12)(13,28,31,37)(14,20,33,18)(16,26,24,27)(17,32,36,30)(22,38,29,40)(25,39,35,34);(1,3,5,2)(4,7,8,6)(9,11)(10,12)(13,29)(14,24)(15,23)(16,33)(17,35)(18,27)(19,21)(20,26)(22,31)(25,36)(28,38)(30,34)(32,39)(37,40);(1,4,5,8)(2,6,3,7)(9,12,15,21)(10,16,19,24)(11,18,23,20)(13,25,31,35)(14,26,33,27)(17,34,36,39)(22,28,29,37)(30,38,32,40);(2,3)(6,7)(9,13)(10,17)(11,22)(12,25)(14,30)(15,31)(16,34)(18,28)(19,36)(20,37)(21,35)(23,29)(24,39)(26,38)(27,40)(32,33);(9,14,15,33)(10,18,19,20)(11,16,23,24)(12,26,21,27)(13,30,31,32)(17,28,36,37)(22,34,29,39)(25,38,35,40)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,e^b=ei,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, 7409033);