#339740

order	512
nilpotency class	3
isoclinism	isoclinic class of #339703
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	#3036	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

582 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)(4,6)(5,9)(7,11)(8,12)(13,14)(15,16)(17,18,21,25)(19,22,27,31)(20,23,28,24)(26,29,32,30);(2,6)(3,11)(4,12)(8,9)(10,14)(13,16)(17,19,21,27)(18,22,25,31)(20,26,28,32)(23,29,24,30);(1,2)(3,7)(4,8)(5,9)(6,12)(10,11)(13,15)(14,16)(18,23)(20,28)(22,29)(24,25)(26,32)(30,31);(1,3,12,16,5,13,4,11)(2,7,6,14,9,15,8,10)(18,23)(20,28)(22,29)(24,25)(26,32)(30,31)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=eh,e^2=h,f^2=i,g^2=h,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=gh,f^c=fi,g^c=gh,f^d=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 246936);