#233683

order	512
nilpotency class	3
isoclinism	isoclinic class of #233668
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

1180 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)(7,17)(8,20)(9,14)(10,15)(11,16)(12,19)(13,18)(21,22,26,23)(24,31,32,36)(25,30,35,28)(27,34,29,33);(1,2)(3,4)(5,7)(6,12)(8,11)(9,18)(10,19)(13,16)(14,17)(15,20)(21,23,26,22)(24,29,32,27)(25,28,35,30)(31,34,36,33);(2,4)(5,8,11,7)(6,13,16,12)(9,18,10,19)(14,17,15,20)(21,24)(22,27)(23,29)(25,34)(26,32)(28,36)(30,31)(33,35);(5,9,11,10)(6,14,16,15)(7,18,8,19)(12,17,13,20)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140879);