#6441583

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1862 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)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,24)(19,25,30,32)(20,26,31,29)(21,28,23,27);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(18,24)(19,26)(20,25)(21,23)(29,30)(31,32);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(17,19,22,30)(18,25,24,32)(20,28,31,27)(21,26,23,29);(1,4,5,3)(2,8,6,7)(17,20,22,31)(18,26,24,29)(19,28,30,27)(21,25,23,32);(17,21,22,23)(18,27,24,28)(19,29,30,26)(20,25,31,32)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6348779);