#6693521

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667484
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	#2541	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1427 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=fgi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi

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, 6600717);