#218859

order	512
nilpotency class	3
isoclinism	isoclinic class of #32706
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	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

669 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)(3,7)(4,6)(5,8)(9,10)(11,12)(13,16)(14,15)(17,18,23,28,22,27,21,26)(19,25,30,32,29,24,20,31);(1,3)(2,5)(4,8)(6,7)(17,19,23,30,22,29,21,20)(18,24,28,31,27,25,26,32);(3,8)(5,7)(9,11,12,10,14,16,13,15)(17,20,21,29,22,30,23,19)(18,25,28,32,27,24,26,31);(2,6)(5,7)(9,12,14,13)(10,11,15,16)(17,21,22,23)(18,26,27,28)(19,20,29,30)(24,32,25,31)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fh,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cei,d^c=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126055);