#231286

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
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

516 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)(9,10,14,11,13,15,12,16)(17,18)(19,26)(20,24)(21,30)(22,25)(23,29)(27,32)(28,31);(2,5)(3,7)(9,11,14,15,13,16,12,10)(19,27)(23,31)(26,32)(28,29);(9,12,13,14)(10,11,15,16)(17,19,20,28,22,27,21,29)(18,23,30,32,25,31,24,26);(1,3,4,7)(2,6,5,8)(9,12,13,14)(10,11,15,16)(17,20,22,21)(18,24,25,30)(19,28,27,29)(23,26,31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fh,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=chi,d^b=de

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, 138482);