#6613053

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	#2847	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1514 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)(4,8)(5,7)(17,18,23,27,22,19,21,25)(20,24,28,31,30,26,29,32);(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)(28,31,29,32);(2,7)(6,8)(10,13)(11,15)(18,19)(24,26)(25,27)(31,32);(1,4,5,3)(2,8,7,6)(9,11)(10,14)(12,15)(13,16)(17,20,22,30)(18,24,19,26)(21,29,23,28)(25,32,27,31);(10,13)(11,15)(17,21,22,23)(18,25,19,27)(20,28,30,29)(24,31,26,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=ghi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6520249);