#343112

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

705 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:

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

Polycyclic presentation:

2^9|a^2=ef,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, 250308);