#7589253

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514196
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25764	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1645 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=fg,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7496449);