#6459233

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438967
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1635 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,23,28,24,30,22,29)(19,31,32,25,20,26,21,27);(2,6)(7,8)(17,19)(18,25)(20,24)(21,23)(22,32)(26,29)(27,30)(28,31);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,22,32,24,19,23,21)(18,26,29,25,30,31,28,27);(1,4,5,3)(2,8,6,7)(17,21,22,20,24,32,23,19)(18,27,29,26,30,25,28,31);(9,11,12,15)(10,14,13,16)(17,22,24,23)(18,28,30,29)(19,32,20,21)(25,31,27,26)

Polycyclic presentation:

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

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, 6366429);