#6844343

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806903
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	#40092	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1452 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=fh,d^2=g,e^2=fgh,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^d=ei

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, 6751539);