#332905

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×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

683 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)(3,8)(4,14)(5,15)(6,16)(7,11)(9,12)(10,13)(17,18,21,26,23,28,22,27)(19,29,30,20,25,32,31,24);(1,3,7,6)(2,8,11,16)(4,12,5,13)(9,15,10,14)(17,18)(19,29)(20,31)(21,27)(22,26)(23,28)(24,30)(25,32);(1,4,7,5)(2,9,11,10)(3,12,6,13)(8,15,16,14)(17,19,23,25)(18,24,28,20)(21,30,22,31)(26,29,27,32);(1,5,7,4)(2,10,11,9)(3,13,6,12)(8,14,16,15)(17,20,23,24)(18,25,28,19)(21,29,22,32)(26,30,27,31)

Polycyclic presentation:

2^9|a^2=e,b^2=hi,c^2=i,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, 240101);