#212367

order	512
nilpotency class	3
isoclinism	isoclinic class of #32884
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

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, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

591 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 119563);