#6678975

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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	#35831	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1756 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6586171);