#6819701

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806527
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#28225	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1431 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)(3,6)(4,8)(5,7)(9,10)(11,14)(12,16)(13,15)(17,18)(19,24)(20,25)(21,23)(22,26)(27,31)(28,32)(29,30);(1,3,5,4)(2,6,7,8)(9,11,14,10)(12,16,13,15)(18,23)(19,27)(20,29)(22,28)(24,25)(30,31);(1,4,5,3)(2,6,7,8)(9,12,14,13)(10,15,11,16)(17,19,28,20,21,29,22,27)(18,24,32,25,23,30,26,31);(9,12,14,13)(10,15,11,16)(18,23)(24,30)(25,31)(26,32);(9,13,14,12)(10,15,11,16)(17,20,22,19,21,27,28,29)(18,25,26,24,23,31,32,30)

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,d^2=fi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=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, 6726897);