#239226

order	512
nilpotency class	3
isoclinism	isoclinic class of #233019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

890 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)(5,6,10,14)(7,16,17,8)(9,15,11,13)(12,19,18,20)(21,22,27,32)(23,24,34,36)(25,30,26,31)(28,29,35,33);(5,7)(6,8)(9,12)(10,17)(11,18)(13,19)(14,16)(15,20)(21,23,27,34)(22,28,32,35)(24,31,36,30)(25,29,26,33);(2,4)(5,8,10,16)(6,12,14,18)(7,13,17,15)(9,19,11,20)(21,24,25,35,27,36,26,28)(22,29,31,23,32,33,30,34);(5,9,10,11)(6,13,14,15)(7,12,17,18)(8,19,16,20)(21,25,27,26)(22,30,32,31)(23,29,34,33)(24,35,36,28)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146422);