#231162

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	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

690 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)(6,8)(9,10,13,15)(11,16,14,12)(17,18,22,27,21,25,28,26)(19,24,30,32,29,23,20,31);(1,2)(3,6)(4,5)(7,8)(17,19,28,20,21,29,22,30)(18,23,26,32,25,24,27,31);(1,3,4,7)(2,6,5,8)(9,11,12,10,13,14,16,15)(17,20,28,29,21,30,22,19)(18,24,27,32,25,23,26,31);(2,5)(6,8)(19,29)(20,30)(23,24)(31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ef,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ci,d^b=de,d^c=di

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, 138358);