#6441393

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1715 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,27)(20,24)(21,31)(22,26)(23,25)(28,32)(29,30);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(19,28)(20,30)(21,23)(24,29)(25,31)(27,32);(10,14)(13,16)(17,19,22,24)(18,20,26,27)(21,28,23,29)(25,32,31,30);(1,4,5,3)(2,6,7,8)(17,20)(18,24)(19,26)(21,30)(22,27)(23,32)(25,28)(29,31);(17,21,22,23)(18,25,26,31)(19,29,24,28)(20,30,27,32)

Polycyclic presentation:

2^9|b^2=fgi,c^2=i,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, 6348589);