#497159

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	C₂⁴	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

536 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=f,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cei

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343785, #72343823, #72343826, …
maximal normal subgroup of	#72343785, #72343823, #72343826, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404355);