#352232

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

Polycyclic presentation:

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

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, 259428);