#6453394

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	#2416	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1499 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)(12,15)(13,16)(17,18)(19,24)(20,25)(21,23)(22,26)(27,30)(28,31)(29,32);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(18,23)(19,27)(20,24)(22,31)(25,29)(30,32);(9,11,13,12)(10,15,16,14)(17,19,21,29)(18,24,23,32)(20,28,30,26)(22,25,31,27);(9,12,13,11)(10,15,16,14)(17,20)(18,25)(19,28)(21,30)(22,32)(23,27)(24,31)(26,29);(1,3,4,7)(2,6,5,8)(18,23)(19,29)(25,27)(26,28)

Polycyclic presentation:

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

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, 6360590);