#6450226

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437510
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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1966 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)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,15,16)(17,18)(19,25)(20,26)(21,23)(22,27)(24,29)(28,31)(30,32);(2,5)(6,8)(18,23)(19,28)(20,22)(24,31)(25,32)(29,30);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,19,21,29)(18,24,23,25)(20,28,22,30)(26,32,27,31);(9,11,12,15)(10,14,13,16)(17,19,21,29)(18,25,23,24)(20,28,22,30)(26,31,27,32);(17,20,21,22)(18,26,23,27)(19,28,29,30)(24,32,25,31)

Polycyclic presentation:

2^9|a^2=gi,c^2=f,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,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, 6357422);