#352086

order	512
nilpotency class	3
isoclinism	isoclinic class of #352081
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

721 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)(5,6,10,8)(7,13,16,20)(9,18,14,12)(11,19,15,17)(23,29)(24,31)(26,32)(28,34)(30,35)(33,36);(5,7,14,19,10,16,9,17)(6,11,18,20,8,15,12,13)(21,22)(23,26)(24,28)(25,27)(29,32)(30,33)(31,34)(35,36);(5,8,10,6)(7,15,16,11)(9,18,14,12)(13,17,20,19)(21,23,24,29,25,30,31,35)(22,26,28,32,27,33,34,36);(2,4)(6,8)(7,16)(12,18)(17,19)(22,27)(26,33)(28,34)(32,36)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,d^b=dhi,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, 259282);