#184039

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄²×D₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

660 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=f,e^2=h,f^2=i,g^2=h,b^a=be,d^a=dhi,e^a=eh,c^b=ci,e^b=eh,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91235);