#6587777

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#35793	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1582 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,23,19)(20,24,32,27)(21,28,30,25)(22,26,31,29);(2,6)(7,8)(17,19,23,18)(20,25,32,28)(21,27,30,24)(22,26,31,29);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20)(18,24)(19,27)(21,31)(22,30)(23,32)(25,26)(28,29);(9,11,12,15)(10,14,13,16)(20,30)(21,32)(22,31)(24,25)(26,29)(27,28);(1,4,5,3)(2,8,6,7)(17,21,23,30)(18,25,19,28)(20,31,32,22)(24,26,27,29)

Polycyclic presentation:

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

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, 6494973);