#7590476

order	512
nilpotency class	3
isoclinism	isoclinic class of #7535045
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3549
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	#59220
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1095 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=g,d^2=f,e^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=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, 7497672);