#6464718

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1506 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)(17,18)(19,29)(20,27)(21,26)(22,25)(23,28)(24,30)(31,32);(9,10)(11,15)(12,13)(14,16)(17,19,22,30,23,31,20,26)(18,24,27,29,28,21,25,32);(2,5)(6,8)(10,13)(14,15)(17,20,23,22)(18,25,28,27)(19,30,31,26)(21,32,24,29);(9,11,12,16)(10,14,13,15)(17,21)(18,26)(19,27)(20,29)(22,32)(23,24)(25,31)(28,30);(1,3,4,7)(2,6,5,8)(17,20,23,22)(18,27,28,25)(19,26,31,30)(21,32,24,29)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,d^c=di,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, 6371914);