#6461480

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23422	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2201 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)(17,18)(19,23)(20,22)(21,25)(24,28)(26,30)(27,29)(31,32);(9,10)(11,15)(12,14)(13,16)(18,22)(20,28)(23,29)(24,25)(27,32)(30,31);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,27)(21,26)(22,29)(24,31)(25,30)(28,32);(1,3,4,7)(2,6,5,8)(10,14)(13,15);(19,26)(23,30)(27,32)(29,31)

Polycyclic presentation:

2^9|c^2=h,d^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=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, 6368676);