#233510

order	512
nilpotency class	3
isoclinism	isoclinic class of #233501
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1212 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)(5,6,11,13)(7,16,17,19)(8,12,20,18)(9,15,10,14)(21,22)(23,30)(24,28)(25,27)(26,32)(29,34)(31,36)(33,35);(2,4)(5,7)(6,8)(9,12)(10,18)(11,17)(13,20)(14,19)(15,16)(21,23,24,32)(22,26,28,30)(25,31,34,33)(27,35,29,36);(5,8,11,20)(6,12,13,18)(7,14,17,15)(9,19,10,16)(22,27)(23,31)(25,34)(28,29)(32,33)(35,36);(6,13)(7,17)(12,18)(14,15)(22,28)(26,30)(27,29)(35,36)

Polycyclic presentation:

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

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, 140706);