#6434101

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432838
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×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1882 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(18,22)(19,26)(20,29)(24,25)(27,28)(30,32);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,18,21,24)(19,23,28,31)(20,25,29,22)(26,30,27,32);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,19,21,28)(18,23,24,31)(20,27,29,26)(22,30,25,32);(10,14)(13,16)(18,24)(22,25)(23,31)(30,32);(17,20,21,29)(18,25,24,22)(19,27,28,26)(23,32,31,30)

Polycyclic presentation:

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

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, 6341297);