#6673329

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667484
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#36529	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2024 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311027, #72311030, #72311041, …
maximal normal subgroup of	#72311027, #72311030, #72311041, …
maximal quotient of	#72311453, #72311584

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6580525);