#7162230

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996671
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	#24214	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1559 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=i,d^2=fgi,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7069426);