#268247

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

970 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:

(5,6,9,11,8,12,15,16)(7,10,14,18,13,19,17,20)(21,22)(23,28,31,35)(24,27,33,34)(25,29)(26,30)(32,36);(1,2)(3,4)(9,15)(11,16)(14,17)(18,20)(21,23,26,31)(22,27,30,34)(24,32,33,25)(28,29,35,36);(2,4)(5,6)(7,10)(8,12)(9,16)(11,15)(13,19)(14,20)(17,18)(23,31)(25,32)(27,34)(29,36);(5,7,8,13)(6,10,12,19)(9,17,15,14)(11,20,16,18)(21,24)(22,28)(23,32)(25,31)(26,33)(27,29)(30,35)(34,36)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=h,e^2=i,f^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,e^c=ehi,f^c=fh,e^d=ei,f^d=fhi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175443);