#346785

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×#3547	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2215 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)(7,16)(8,13)(9,14)(10,19)(11,15)(12,17)(18,20)(21,22,25,29)(23,30,32,35)(24,28,26,33)(27,34,36,31);(5,7,11,8)(6,12,15,20)(9,17,10,18)(13,19,16,14)(21,22)(23,27)(24,28)(25,29)(26,33)(30,34)(31,35)(32,36);(5,8,11,7)(6,13,15,16)(9,17,10,18)(12,19,20,14)(21,23)(22,27)(24,31)(25,32)(26,34)(28,35)(29,36)(30,33);(2,4)(5,9,11,10)(6,14,15,19)(7,17,8,18)(12,13,20,16)(21,24,25,26)(22,28,29,33)(23,31,32,34)(27,35,36,30)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=hi,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, 253981);