#7590479

order	512
nilpotency class	3
isoclinism	isoclinic class of #7535048
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

1085 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:

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

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,d^c=di,e^c=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, 7497675);