#233659

order	512
nilpotency class	3
isoclinism	isoclinic class of #233642
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

1260 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140855);