#140841

order	512
nilpotency class	4
isoclinism	isoclinic class of #2212
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#459
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9674
prime radical	#3147	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

406 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=e,d^2=gi,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=dg,f^b=fg,g^b=gi,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225426, #72225428, #72225431, …
maximal normal subgroup of	#72225426, #72225428, #72225431, …
maximal quotient of	#72227201

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48037);