#231373

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
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, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

533 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=fh,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ch,d^b=de,d^c=dhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 138569);