#7590480

order	512
nilpotency class	3
isoclinism	isoclinic class of #7535047
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:

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

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^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, 7497676);