#212059

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
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	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

623 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)(4,5)(7,8)(9,10,14,16,13,12,11,15)(17,18,20,24)(19,26,27,22)(21,23,29,25)(28,31,32,30);(2,5)(6,8)(10,12)(15,16)(17,19,20,27)(18,22,24,26)(21,28,29,32)(23,30,25,31);(1,3,4,7)(2,6,5,8)(9,11,13,14)(10,15,12,16)(18,23)(21,29)(22,30)(24,25)(26,31)(28,32);(9,12,14,15,13,10,11,16)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=eh,d^2=gh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=di,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, 119255);