#346772

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×#3552	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2187 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)(7,11)(8,10)(9,13)(12,16)(14,18)(15,17)(19,20)(21,22,26,23)(24,31,32,36)(25,30,35,28)(27,34,29,33);(6,10)(8,16)(11,17)(12,13)(15,20)(18,19)(21,23,26,22)(24,29,32,27)(25,28,35,30)(31,34,36,33);(5,7)(6,11)(8,15)(9,14)(10,17)(12,19)(13,18)(16,20)(21,24)(22,27)(23,29)(25,34)(26,32)(28,36)(30,31)(33,35);(2,4)(7,14)(11,18)(15,20)(17,19)

Polycyclic presentation:

2^9|a^2=i,b^2=i,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, 253968);