#6441105

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×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1797 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,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, 6348301);