#6456788

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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	#35234	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1577 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)(19,25)(20,26)(21,30)(22,24)(23,27)(28,32)(29,31);(1,3,5,4)(2,6,8,7)(18,24)(19,28)(20,29)(21,23)(25,26)(31,32);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,19)(18,25)(20,23)(21,28)(22,29)(24,31)(26,27)(30,32);(1,4,5,3)(2,6,8,7)(17,20,22,28)(18,26,24,32)(19,23,29,21)(25,27,31,30);(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,27,24,30)(19,28,29,20)(25,26,31,32)

Polycyclic presentation:

2^9|b^2=fi,c^2=fi,d^2=f,e^2=gi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,e^b=ei,h^b=hi,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, 6363984);