#7501357

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	#60456	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2128 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)(3,4)(5,7)(6,8)(9,10,15,20)(11,16,24,33)(12,25,29,36)(13,30,14,19)(17,27,37,28)(18,22,32,23)(21,35,38,34)(26,31,39,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,26)(13,23)(14,22)(15,24)(17,34)(18,30)(19,32)(20,33)(21,27)(25,31)(28,38)(29,39)(35,37)(36,40);(1,4,5,6)(2,3,7,8)(9,12)(10,17)(11,21)(13,27)(14,28)(15,29)(16,31)(18,35)(19,36)(20,37)(22,26)(23,39)(24,38)(25,30)(32,34)(33,40);(2,7)(3,8)(9,11)(10,18)(12,21)(13,22)(14,23)(15,24)(16,30)(17,35)(19,33)(20,32)(25,31)(26,27)(28,39)(29,38)(34,37)(36,40);(9,13,15,14)(10,19,20,30)(11,22,24,23)(12,27,29,28)(16,18,33,32)(17,36,37,25)(21,26,38,39)(31,35,40,34)

Polycyclic presentation:

2^9|a^2=i,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, 7408553);