#6537219

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	#2011	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1547 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)(3,8)(4,6)(5,7)(10,13)(14,16)(17,18)(19,22)(20,28)(21,25)(23,27)(24,29)(26,31)(30,32);(1,3,5,4)(2,6,7,8)(9,10,12,13)(11,14,15,16)(17,19,22,18)(20,26,30,24)(21,25,23,27)(28,29,32,31);(10,13)(14,16)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28);(1,4,5,3)(2,6,7,8)(11,15)(14,16);(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,25,19,27)(20,29,30,31)(24,32,26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6444415);