#6437007

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432833
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#30105	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1708 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)(10,13)(11,14)(15,16)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32);(2,5)(6,8)(9,10,12,13)(11,15,16,14)(17,19)(18,22)(20,26)(21,25)(23,28)(24,29)(27,31)(30,32);(9,11)(10,14)(12,16)(13,15)(19,25)(22,28)(26,31)(29,32);(1,3,4,7)(2,6,5,8)(9,10,12,13)(11,14,16,15)(18,23)(22,28)(24,30)(29,32);(9,10,12,13)(11,14,16,15)(17,20,21,27)(18,24,23,30)(19,26,25,31)(22,29,28,32)

Polycyclic presentation:

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

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, 6344203);