#6547131

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535696
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2018	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1621 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,g^c=gi,e^d=eh

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, 6454327);