#6449407

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	#35564	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1948 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)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,21)(19,24,28,30)(20,29,23,25)(26,31,32,27);(2,5)(6,8)(17,19)(18,24)(20,27)(21,30)(22,28)(23,31)(25,32)(26,29);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20)(18,25)(21,29)(22,23)(24,30)(27,31);(9,11,12,15)(10,14,13,16)(17,21,22,18)(19,26,28,32)(20,29,23,25)(24,31,30,27);(17,20,22,23)(18,25,21,29)(19,27,28,31)(24,32,30,26)

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=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, 6356603);