#7525826

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	#23871	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1712 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)(9,10,12,14)(11,13,15,16)(18,23)(24,30)(25,26)(31,32);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,18,21,23)(19,24,28,30)(20,26,22,25)(27,32,29,31);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,19,21,28)(18,24,23,30)(20,27,22,29)(25,31,26,32);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(18,25)(20,22)(23,26)(24,31)(27,29)(30,32);(2,6)(7,8)(17,20,21,22)(18,26,23,25)(19,27,28,29)(24,32,30,31)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=f,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^d=ei,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, 7433022);