#267120

order	512
nilpotency class	3
isoclinism	isoclinic class of #267051
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	C₂×#3106	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1064 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)(5,6,10,18,9,13,8,12)(7,14,15,11,17,19,16,20)(21,22,26,28,24,29,25,30)(23,31,34,35,33,27,32,36);(2,4)(21,23,24,33)(22,27,29,31)(25,32,26,34)(28,36,30,35);(5,7)(6,11)(8,15)(9,17)(10,16)(12,19)(13,20)(14,18)(22,28)(23,32)(25,26)(29,30)(33,34)(35,36);(2,4)(5,8,9,10)(6,12,13,18)(7,15,17,16)(11,19,20,14)(22,29)(27,31)(28,30)(35,36)

Polycyclic presentation:

2^9|a^2=fh,b^2=i,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=gi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 174316);