#7569726

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	#36434	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1801 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)(17,18,21,25)(19,26,29,23)(20,24,31,22)(27,28,32,30);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(17,19,21,29)(18,23,25,26)(20,27,31,32)(22,28,24,30);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20)(18,24)(19,27)(21,31)(22,25)(23,30)(26,28)(29,32);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(19,28)(20,31)(22,24)(23,27)(26,32)(29,30);(2,6)(7,8)(10,14)(13,15)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=fgi,d^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,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, 7476922);