#332910

order	512
nilpotency class	3
isoclinism	isoclinic class of #332839
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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

666 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240106);