#6488806

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25733	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1709 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=h,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,d^c=di,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, 6396002);