#197638

order	512
nilpotency class	3
isoclinism	isoclinic class of #19162
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#60536	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=hi,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322077, #72322165, #72322177
maximal normal subgroup of	#72322077, #72322165, #72322177

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 104834);