#6700784

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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	#25174	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1542 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)(9,10)(11,14)(12,13)(15,16)(19,25)(21,30)(27,32)(28,29);(1,3,4,8)(2,5,6,7)(17,18,22,26)(19,27,25,32)(20,24,23,31)(21,28,30,29);(2,6)(5,7)(10,13)(14,16)(17,19,22,25)(18,21,26,30)(20,29,23,28)(24,32,31,27);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20,22,23)(18,24,26,31)(19,28,25,29)(21,27,30,32);(3,8)(5,7)(17,21)(18,25)(19,26)(20,27)(22,30)(23,32)(24,29)(28,31)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6607980);