#6508929

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
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	#37273	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1957 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)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,28,25)(23,31,34,35)(24,26,30,27)(29,36,32,33);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,33,28,34,26,36)(22,29,30,31,25,32,24,35);(2,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24)(22,27)(23,31)(25,26)(28,30)(29,36)(32,33)(34,35);(8,16)(11,20)(13,15)(17,19)(21,25,28,22)(23,32,34,29)(24,26,30,27)(31,33,35,36);(7,14)(10,18)(13,15)(17,19)(21,26,28,27)(22,30,25,24)(23,33,34,36)(29,35,32,31)

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6416125);