#6696054

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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	#25762	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1655 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6603250);