#6698205

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667836
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	#24435	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1588 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)(3,6)(4,5)(7,8)(10,14)(11,12)(17,18,21,23)(19,24,28,30)(20,25,27,31)(22,26,29,32);(9,10,13,14)(11,15,12,16)(18,23)(19,27)(20,28)(22,29)(24,25)(30,31);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,19,21,28)(18,24,23,30)(20,29,27,22)(25,32,31,26);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(18,23)(19,28)(20,27)(26,32);(10,14)(15,16)(17,20,21,27)(18,25,23,31)(19,22,28,29)(24,26,30,32)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,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, 6605401);