#6555764

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	#24439	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1582 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)(3,8,4,6)(17,18,22,28,24,19,23,26)(20,29,30,25,27,32,31,21);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,24,18)(20,25,27,21)(22,28,23,26)(29,31,32,30);(10,13)(14,16)(17,20,24,27)(18,25,19,21)(22,31,23,30)(26,32,28,29);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(17,21)(18,20)(19,27)(22,29)(23,32)(24,25)(26,31)(28,30);(11,15)(14,16)(17,22,24,23)(18,26,19,28)(20,30,27,31)(21,32,25,29)

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=i,d^2=fgi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,f^b=fi,g^b=gi,e^c=ei,f^c=fi,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, 6462960);