#6720566

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	#42909	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1743 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,14)(13,15)(17,18,22,26)(19,21,25,31)(20,30,23,24)(27,28,32,29);(1,3)(2,5)(4,8)(6,7)(17,18)(19,27)(20,24)(21,29)(22,26)(23,30)(25,32)(28,31);(9,11,13,12)(10,14,15,16)(17,19)(18,21)(20,29)(22,25)(23,28)(24,32)(26,31)(27,30);(2,6)(5,7)(17,20,22,23)(18,24,26,30)(19,28,25,29)(21,27,31,32);(3,8)(5,7)(9,12,13,11)(10,14,15,16)(17,21)(18,25)(19,26)(20,27)(22,31)(23,32)(24,29)(28,30)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi

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, 6627762);