#358549

order	512
nilpotency class	3
isoclinism	isoclinic class of #8736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#381
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#381
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1646 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)(7,12,18,26,19,27,24,30)(8,20,23,28,22,13,21,29)(17,25,31,35,32,33,34,36);(1,2)(3,4)(5,7)(6,12,15,27)(8,17)(9,18)(10,24)(11,19)(13,25,20,33)(14,26,16,30)(21,31)(22,32)(23,34)(28,35,29,36);(2,4)(5,8)(6,13,15,20)(7,17)(9,21)(10,23)(11,22)(12,25,27,33)(14,28,16,29)(18,31)(19,32)(24,34)(26,35,30,36);(5,9,11,10)(6,14,15,16)(7,18,19,24)(8,21,22,23)(12,26,27,30)(13,28,20,29)(17,31,32,34)(25,35,33,36)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=ef,d^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,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, 265745);