#358293

order	512
nilpotency class	3
isoclinism	isoclinic class of #2032
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
commutator	C₂⁴	C₂³×C₄
Frattini, prime radical, cosocle	C₂⁵	C₂⁴
socle	C₂²	C₂×#393
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1726 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)(7,15,18,11)(8,19)(9,13)(10,14)(12,22)(16,24,20,26)(17,29,31,25)(21,32)(23,28)(27,34)(30,35,33,36);(1,2)(3,4)(5,7)(6,11)(8,20,22,16)(9,17)(10,18)(12,26,19,24)(13,25)(14,15)(21,33,34,30)(23,31)(27,36,32,35)(28,29);(2,4)(5,8)(6,12,14,19)(7,16)(9,21)(10,22)(11,24,15,26)(13,27,28,32)(17,30)(18,20)(23,34)(25,35,29,36)(31,33);(5,9,10,23)(6,13,14,28)(7,17,18,31)(8,21,22,34)(11,25,15,29)(12,27,19,32)(16,30,20,33)(24,35,26,36)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,d^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,g^b=gi,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, 265489);