#7590481

order	512
nilpotency class	3
isoclinism	isoclinic class of #7535046
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

1078 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:

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

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,e^c=ei,e^d=ei

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, 7497677);