#6528780

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499780
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24129	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1523 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26,30,33)(27,34,37,39)(28,32,29,38)(31,36,40,35);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,40)(34,38);(2,5)(6,8)(11,17)(12,19)(14,20)(15,22)(25,28,30,29)(26,32,33,38)(27,35,37,36)(31,39,40,34);(9,12,13,19)(10,15,16,22)(11,18,17,23)(14,21,20,24)(25,28,30,29)(26,32,33,38)(27,36,37,35)(31,34,40,39);(1,3,4,7)(2,6,5,8)(11,17)(14,20)(18,23)(21,24)(27,37)(31,40)(34,39)(35,36)

Polycyclic presentation:

2^9|a^2=hi,c^2=i,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ehi,f^b=fi,d^c=dh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6435976);