#7535049

order	512
nilpotency class	3
isoclinism	isoclinic class of #7535047
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3549
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	#59041
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1754 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)(3,15)(4,21)(5,24)(6,13)(7,12)(8,14)(9,19)(10,22)(11,23)(16,28)(17,30)(18,26)(20,31)(25,29)(27,32);(1,3,6,19,8,20,7,25)(2,9,12,15,14,29,13,31)(4,17,5,18,23,32,22,16)(10,30,11,28,24,27,21,26);(1,4,7,22,8,23,6,5)(2,10,13,21,14,24,12,11)(3,16,19,17,20,18,25,32)(9,26,15,30,29,28,31,27);(1,4,6,5,8,23,7,22)(2,10,12,11,14,24,13,21)(3,17,19,18,20,32,25,16)(9,27,15,26,29,30,31,28);(1,5,6,23,8,22,7,4)(2,11,12,24,14,21,13,10)(3,18,19,32,20,16,25,17)(9,28,15,27,29,26,31,30)

Polycyclic presentation:

2^9|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, 7442245);