#268539

order	512
nilpotency class	3
isoclinism	isoclinic class of #268491
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
prime radical	#59033	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1101 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=h,e^2=i,f^2=hi,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=fhi,e^d=ei,f^d=fhi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175735);