#6466747

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1457 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)(11,14)(12,16)(13,15)(19,26)(23,31)(27,30)(29,32);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,18)(19,27)(20,21)(22,24)(23,29)(25,28)(26,30)(31,32);(9,12,13,11)(10,14,15,16)(17,19,28,32,22,26,21,29)(18,23,20,30,24,31,25,27);(17,20,28,24,22,25,21,18)(19,27,32,23,26,30,29,31);(1,3,4,7)(2,6,5,8)(17,21,22,28)(18,20,24,25)(19,29,26,32)(23,30,31,27)

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=hi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,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, 6373943);