#6569472

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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	#2537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1386 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,c^2=f,d^2=fghi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ei,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6476668);