#234038

order	512
nilpotency class	3
isoclinism	isoclinic class of #234028
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1268 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=cg,d^b=di,e^b=eh,d^c=dh,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141234);