#346760

order	512
nilpotency class	3
isoclinism	isoclinic class of #346740
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2168 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^c=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, 253956);