#6842018

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806903
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	#40053	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1498 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=fhi,d^2=fg,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6749214);