#6468220

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1672 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)(11,15)(12,16)(13,14)(17,18,21,24,22,25,20,29)(19,23,28,30,27,31,26,32);(10,14)(11,12)(17,18)(19,23)(20,24)(21,29)(22,25)(26,30)(27,31)(28,32);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,19)(18,23)(20,26)(21,28)(22,27)(24,30)(25,31)(29,32);(9,12,13,11)(10,16,14,15)(17,20,22,21)(18,24,25,29)(19,26,27,28)(23,30,31,32);(1,3,4,7)(2,6,5,8)(19,27)(23,31)(26,28)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375416);