#6482198

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470811
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	C₂×#3251	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1648 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)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,18,23,26)(19,21,25,22)(20,28,31,24)(27,30,32,29);(1,3)(2,6)(4,11)(5,9)(7,15)(8,13)(10,12)(14,16)(17,19,22,18,23,25,21,26)(20,27,30,24,31,32,29,28);(3,9)(6,13)(10,11)(14,15)(17,20,22,30,23,31,21,29)(18,24,25,32,26,28,19,27);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15);(2,8)(6,13)(7,16)(14,15)(17,21,23,22)(18,25,26,19)(20,29,31,30)(24,32,28,27)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,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, 6389394);