#6434218

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	#23428	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1946 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=fg,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,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, 6341414);