#314254

order	512
nilpotency class	3
isoclinism	isoclinic class of #264427
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

760 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)(3,6)(4,5)(7,8)(17,18,22,26)(19,27,29,24)(20,23,30,32)(21,31,28,25);(2,5)(6,8)(9,10)(11,13)(12,15)(14,16)(17,19,21,20,22,29,28,30)(18,23,25,27,26,32,31,24);(1,3,4,7)(2,6,5,8)(10,13)(11,16)(14,15)(17,20,28,19,22,30,21,29)(18,24,31,32,26,27,25,23);(2,5)(6,8)(17,19,28,30,22,29,21,20)(18,23,31,24,26,32,25,27)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=eh,d^2=fh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=dh,g^b=gi,d^c=dh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 221450);