#339871

order	512
nilpotency class	3
isoclinism	isoclinic class of #339847
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3253	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

525 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 247067);