#339770

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

550 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^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, 246966);