#7535050

order	512
nilpotency class	3
isoclinism	isoclinic class of #7535046
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)(3,13)(4,19)(5,9)(6,22)(7,12)(8,16)(10,20)(11,23)(14,24)(15,25)(17,28)(18,27)(21,30)(26,32)(29,31);(1,3,6,16,7,18,23,17)(2,8,11,13,12,28,22,27)(4,15,5,14,20,31,21,32)(9,29,19,24,30,25,10,26);(1,4,6,5,7,20,23,21)(2,9,11,19,12,30,22,10)(3,14,17,15,18,32,16,31)(8,24,27,29,28,26,13,25);(1,4,6,5,7,20,23,21)(2,9,11,19,12,30,22,10)(3,15,16,14,18,31,17,32)(8,25,13,26,28,29,27,24);(1,5,6,20,7,21,23,4)(2,10,11,9,12,19,22,30)(3,14,16,31,18,32,17,15)(8,26,13,29,28,24,27,25)

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,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, 7442246);