#212440

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	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

637 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)(3,8)(4,6)(5,7)(9,10,14,15,13,16,11,12)(17,18,22,26)(19,23,28,32)(20,25,21,24)(27,30,29,31);(1,3,5,4)(2,6,7,8)(9,11,13,14)(10,15,16,12)(17,19,22,28)(18,23,26,32)(20,29,21,27)(24,30,25,31);(10,16)(12,15)(18,24)(20,21)(23,30)(25,26)(27,29)(31,32);(1,4,5,3)(2,6,7,8)(9,12,11,16,13,15,14,10)(17,20,22,21)(18,25,26,24)(19,27,28,29)(23,30,32,31)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=eg,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=dhi,d^c=dhi,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, 119636);