#7503663

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂×D₄²
prime radical	#60422	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1925 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)(9,10,14,20,15,21,16,22)(11,23,26,40,27,17,37,38)(12,28,32,39,31,25,30,18)(13,33,36,19,35,24,34,29);(1,3,5,2)(4,7,8,6)(9,11,15,27)(10,17,21,23)(12,29,31,19)(13,28,35,25)(14,26,16,37)(18,36,39,34)(20,38,22,40)(24,32,33,30);(1,4,5,8)(2,6,3,7)(9,12,15,31)(10,18,21,39)(11,24,27,33)(13,23,35,17)(14,30,16,32)(19,37,29,26)(20,25,22,28)(34,40,36,38);(2,3)(6,7)(9,13)(10,19)(11,25)(12,23)(14,34)(15,35)(16,36)(17,31)(18,37)(20,33)(21,29)(22,24)(26,39)(27,28)(30,40)(32,38);(9,14,15,16)(10,20,21,22)(11,26,27,37)(12,30,31,32)(13,34,35,36)(17,38,23,40)(18,25,39,28)(19,33,29,24)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7410859);