#219171

order	512
nilpotency class	3
isoclinism	isoclinic class of #32706
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

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 ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

581 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=f,d^2=e,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, 126367);