#343130

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	#18687	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

657 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:

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

Polycyclic presentation:

2^9|a^2=efhi,c^2=i,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, 250326);