#7576386

order	512
nilpotency class	3
isoclinism	isoclinic class of #7513980
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	#23858	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1738 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)(3,7)(4,6)(5,8)(10,14)(11,12)(17,18,22,20)(19,23,28,25)(21,24,30,29)(26,31,27,32);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,16,12,15)(17,19,22,28)(18,23,20,25)(21,26,30,27)(24,31,29,32);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,14,16)(17,20,22,18)(19,25,28,23)(21,29,30,24)(26,32,27,31);(1,4,5,3)(2,6,8,7)(9,11,13,12)(10,16,14,15)(19,26)(21,30)(23,31)(24,29)(25,32)(27,28);(2,8)(6,7)(9,12,13,11)(10,15,14,16)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fi,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,h^d=hi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7483582);