#6472801

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469956
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	#35251	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2503 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6379997);