#333562

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

726 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,9,13,10,14,20,16)(7,15,12,8,17,18,19,11)(21,22,25,28,26,24,35,29)(23,30,33,31,34,36,32,27);(1,2)(3,4)(5,7,10,17)(6,11,14,8)(9,19,20,12)(13,18,16,15)(21,23,26,34)(22,27,24,31)(25,32,35,33)(28,36,29,30);(2,4)(21,24)(22,26)(23,31)(25,29)(27,34)(28,35)(30,32)(33,36);(5,8,10,11)(6,12,14,19)(7,16,17,13)(9,15,20,18)(23,32)(25,35)(27,36)(28,29)(30,31)(33,34)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240758);