#196260

order	512
nilpotency class	3
isoclinism	isoclinic class of #196257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1183 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)(3,9)(4,12,14,7)(5,11)(6,8)(10,13,16,15)(17,18,20,24,21,23,27,30)(19,22,26,29,25,28,31,32);(1,3,9,2,6,11,5,8)(4,10,13,7,14,16,15,12)(17,19,20,26,21,25,27,31)(18,22,24,29,23,28,30,32);(1,4,6,14)(2,7,8,12)(3,10,11,16)(5,15,9,13)(19,25)(22,28)(26,31)(29,32);(18,23)(22,28)(24,30)(29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239303, #72257893, #72257895
maximal normal subgroup of	#72239303, #72257893, #72257895
maximal quotient of	#72239308

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103456);