#357891

order	512
nilpotency class	3
isoclinism	isoclinic class of #9109
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2538 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^b=hi,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265087);