#352079

order	512
nilpotency class	3
isoclinism	isoclinic class of #352064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

729 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)(6,10)(8,11)(12,16)(13,17)(15,18)(19,20)(21,22,26,30)(23,31,33,28)(24,27,34,36)(25,35,32,29);(5,6,11,17,9,13,8,10)(7,12,18,20,14,19,15,16)(21,23,32,34,26,33,25,24)(22,27,35,28,30,36,29,31);(5,7,9,14)(6,12,13,19)(8,15,11,18)(10,16,17,20)(21,24,25,33,26,34,32,23)(22,28,29,27,30,31,35,36);(2,4)(7,14)(12,19)(15,18)(16,20)(23,33)(24,34)(27,36)(28,31)

Polycyclic presentation:

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

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, 259275);