#343134

order	512
nilpotency class	3
isoclinism	isoclinic class of #343112
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2679
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

613 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 250330);