#7532110

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	#59080	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1826 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16);(2,5)(6,8)(11,16)(13,15)(17,18)(19,26)(20,24)(21,29)(22,25)(23,27)(28,31)(30,32);(9,11,12,16)(10,13,14,15);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,19,22,28)(18,23,25,32)(20,27,21,30)(24,31,29,26);(2,5)(6,8)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)

Polycyclic presentation:

2^9|c^2=g,d^2=fi,e^2=i,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, 7439306);