#226894

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

612 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,d^b=de,e^b=eh,g^b=gh,d^c=dhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 134090);