#6588933

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573485
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1522 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)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,24,19)(20,25,22,27)(21,26,23,28)(29,30,32,31);(1,3,5,4)(2,6,8,7)(17,19,24,18)(20,29,22,32)(21,26,23,28)(25,30,27,31);(10,13)(14,16)(17,20,24,22)(18,25,19,27)(21,30,23,31)(26,32,28,29);(1,3,5,4)(2,7,8,6)(9,11,12,15)(10,14,13,16)(17,21)(18,26)(19,28)(20,22)(23,24)(25,27);(1,4,5,3)(2,6,8,7)(17,22,24,20)(18,27,19,25)(21,31,23,30)(26,29,28,32)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=fg,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dhi,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, 6496129);