#196255

order	512
nilpotency class	3
isoclinism	isoclinic class of #196248
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2433	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1373 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=hi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,f^b=fh,e^c=eh,g^c=gh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239301, #72257616, #72257620
maximal normal subgroup of	#72239301, #72257616, #72257620
maximal quotient of	#72239310

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103451);