#6540224

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	#23973	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1714 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)(3,7)(4,6)(5,8)(17,18,23,19)(20,28,30,32)(21,25,22,27)(24,31,26,29);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,18)(20,26,30,24)(21,25,22,27)(28,29,32,31);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,20,23,30)(18,24,19,26)(21,29,22,31)(25,32,27,28);(1,4,5,3)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(17,21,23,22)(18,25,19,27)(20,29,30,31)(24,32,26,28);(11,15)(14,16)(20,30)(24,26)(28,32)(29,31)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fi,d^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^c=ei,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, 6447420);