#7503842

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500908
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	#60426	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2112 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=f,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,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, 7411038);