#6447308

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×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1902 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=fg,d^2=fi,h^2=i,b^a=bf,c^a=cg,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, 6354504);