#333586

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

720 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,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, 240782);