#343019

order	512
nilpotency class	3
isoclinism	isoclinic class of #343003
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	#18678	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

713 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=hi,c^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=def,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 250215);