#267272

order	512
nilpotency class	3
isoclinism	isoclinic class of #267211
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

1014 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)(5,6,10,12,9,14,8,13)(7,15,16,11,18,20,17,19)(21,22,24,29,25,28,33,27)(23,26,30,35,32,34,31,36);(2,4)(21,23)(22,26)(24,30)(25,32)(27,36)(28,34)(29,35)(31,33);(5,7)(6,11)(8,16)(9,18)(10,17)(12,15)(13,20)(14,19)(22,27)(23,30)(24,33)(28,29)(31,32)(35,36);(2,4)(5,8,9,10)(6,12,14,13)(7,16,18,17)(11,15,19,20)(22,28)(26,34)(27,29)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 174468);