#334624

order	512
nilpotency class	3
isoclinism	isoclinic class of #334567
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#535
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

616 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=hi,c^2=i,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=gi,f^c=fi,g^c=gi,e^d=eh,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, 241820);