#334073

order	512
nilpotency class	3
isoclinism	isoclinic class of #332071
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	C₂×#547	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

769 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=defh,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

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 241269);