#352053

order	512
nilpotency class	3
isoclinism	isoclinic class of #352047
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

766 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)(7,13)(8,10)(9,17)(11,15)(12,16)(14,19)(18,20)(21,22,26,28)(23,29,31,36)(24,25,32,33)(27,34,35,30);(6,10)(11,18)(12,17)(13,19)(21,23,25,30,26,31,33,34)(22,27,24,29,28,35,32,36);(5,7,9,15,8,14,16,20)(6,11,12,13,10,18,17,19)(21,24,25,28,26,32,33,22)(23,29,30,35,31,36,34,27);(2,4)(22,28)(24,32)(27,35)(29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259249);