#334792

order	512
nilpotency class	3
isoclinism	isoclinic class of #334695
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3036	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

643 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 241988);