#6673310

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667460
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	#23963	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2000 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)(21,22,25,28)(23,26,31,35)(24,27,32,36)(29,33,30,34);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23,25,31)(22,26,28,35)(24,29,32,30)(27,33,36,34);(2,4)(23,29)(24,32)(26,33)(27,36)(30,31)(34,35);(6,11)(10,13)(12,19)(18,20);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(23,29)(24,32)(26,33)(27,36)(30,31)(34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284360, #72284514, #72285360, …
maximal normal subgroup of	#72284360, #72284514, #72285360, …
maximal quotient of	#72311453, #72311805

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6580506);