#6556607

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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	#2416	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1409 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)(3,7)(4,8)(5,6)(17,18,23,25)(19,26,27,21)(20,22,29,30)(24,31,32,28);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,28,23,27,30,31)(18,24,20,26,25,32,29,21);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,22,25,23,29,30,18)(19,21,28,24,27,26,31,32);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,15)(13,16)(17,21,23,26)(18,19,25,27)(20,28,29,31)(22,24,30,32);(11,15)(14,16)(19,27)(21,26)(24,32)(28,31)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,e^d=eh

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, 6463803);