#240520

order	512
nilpotency class	3
isoclinism	isoclinic class of #233184
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×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

899 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=e,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=dhi,e^b=eh,f^b=fh,g^b=gh,d^c=di,f^c=fhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 147716);