#343022

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	#18680	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:

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

Polycyclic presentation:

2^9|a^2=efi,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, 250218);