#6731029

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#30593	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1373 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)(17,18,22,23)(19,24,27,32)(20,28,29,25)(21,26,31,30);(10,13)(11,15)(17,19)(18,24)(20,26)(21,25)(22,27)(23,32)(28,31)(29,30);(2,6)(7,8)(9,11,12,15)(10,14,13,16)(17,20)(18,25)(19,21)(22,29)(23,28)(24,30)(26,32)(27,31);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,18,22,23)(19,24,27,32)(20,25,29,28)(21,30,31,26);(1,4,5,3)(2,8,6,7)(17,21)(18,26)(19,20)(22,31)(23,30)(24,28)(25,32)(27,29)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6638225);