#293674

order	512
nilpotency class	3
isoclinism	isoclinic class of #271515
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

854 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)(4,5)(6,8)(9,10)(11,14)(12,16)(13,15)(17,18,20,22)(19,25,27,32)(21,29,26,24)(23,31,30,28);(2,5)(6,7)(9,10,13,15)(11,14,16,12)(18,22)(23,30)(24,29)(25,32);(9,11,12,15,13,16,14,10)(17,19,26,31,20,27,21,28)(18,23,29,32,22,30,24,25);(1,3,4,8)(2,6,5,7)(9,12,13,14)(10,11,15,16)(18,22)(19,27)(24,29)(28,31)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=fh,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^b=gi,d^c=dhi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 200870);