#6483080

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470884
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	#2856	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1480 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)(11,15)(12,16)(13,14)(17,18,23,21)(19,25,28,30)(20,29,27,24)(22,31,32,26);(2,5)(6,8)(10,14)(11,12)(17,19,22,27,23,28,32,20)(18,24,26,25,21,29,31,30);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,20,22,19,23,27,32,28)(18,25,26,29,21,30,31,24);(9,12,13,11)(10,16,14,15)(17,21,23,18)(19,24,28,29)(20,30,27,25)(22,31,32,26);(10,14)(15,16)(18,21)(24,29)(25,30)(26,31)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,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, 6390276);