#6496554

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470415
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	#60561	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1720 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:

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

Polycyclic presentation:

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

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, 6403750);