#6590957

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1590 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:

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

Polycyclic presentation:

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

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, 6498153);