#501264

order	512
nilpotency class	4
isoclinism	isoclinic class of #501262
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61049	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1140 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,e^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=ehi,f^b=fi,h^b=hi,e^c=ei,f^c=fi,e^d=ehi,f^d=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252745
maximal normal subgroup of	#72252745

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408460);