#6538345

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	#37648	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1943 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)(3,6)(4,5)(7,8)(17,18)(19,27)(20,24)(21,25)(22,26)(23,28)(29,31)(30,32);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,25)(21,24)(22,26)(27,28)(29,32)(30,31);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,19,22,29)(18,23,26,32)(20,28,21,30)(24,31,25,27);(9,11)(10,14)(12,15)(13,16)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,31,32,27);(11,15)(14,16)(17,20,22,21)(18,25,26,24)(19,28,29,30)(23,27,32,31)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6445541);