#6700721

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1448 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)(17,18,22,26)(19,24,29,21)(20,31,23,25)(27,28,32,30);(1,3,4,8)(2,5,6,7)(17,18)(19,27)(20,25)(21,28)(22,26)(23,31)(24,30)(29,32);(2,6)(5,7)(10,13)(14,16)(17,19,22,29)(18,24,26,21)(20,30,23,28)(25,32,31,27);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20,22,23)(18,25,26,31)(19,28,29,30)(21,32,24,27);(3,8)(5,7)(17,21,22,24)(18,19,26,29)(20,32,23,27)(25,28,31,30)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=fg,e^2=i,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, 6607917);