#111189

order	512
nilpotency class	3
isoclinism	isoclinic class of #9239
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

632 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,d^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,d^b=dh,e^b=ei,f^b=fi,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18385);