#6617148

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	C₂×#447	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1561 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)(4,7)(5,8)(17,18,22,23,21,25,20,26)(19,24,29,30,27,31,28,32);(1,3,5,4)(2,6,8,7)(9,10,12,13)(11,14,15,16)(18,23)(20,22)(24,30)(25,26)(28,29)(31,32);(1,3,5,4)(2,7,8,6)(10,13)(11,15)(17,19)(18,24)(20,28)(21,27)(22,29)(23,30)(25,31)(26,32);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(19,27)(24,31)(28,29)(30,32);(1,4,5,3)(2,7,8,6)(10,13)(11,15)(18,25)(23,26)(24,31)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6524344);