#6446958

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	#23386	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2102 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,16)(13,15)(18,22)(23,31)(24,25)(26,32);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,14,15,16)(17,18,20,22)(19,26,27,32)(21,24,29,25)(23,30,31,28);(9,12,13,11)(10,14,15,16)(17,19)(18,23)(20,27)(21,30)(22,31)(24,26)(25,32)(28,29);(1,4,5,3)(2,6,7,8)(18,24)(21,29)(22,25)(23,26)(28,30)(31,32);(19,27)(23,31)(26,32)(28,30)

Polycyclic presentation:

2^9|b^2=fg,c^2=g,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, 6354154);