#8759877

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#35072	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4307 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(9,10)(11,15)(12,19)(13,14)(16,21)(17,24)(18,22)(20,23)(25,26,29,32)(27,30,34,38)(28,35,36,31)(33,39,40,37);(1,2)(3,7)(4,5)(6,8)(10,14)(15,22)(16,19)(23,24)(25,27,29,34)(26,30,32,38)(28,33,36,40)(31,37,35,39);(2,5)(6,7)(9,11,13,18)(10,15,14,22)(12,20,21,17)(16,24,19,23)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(1,2)(3,6)(4,5)(7,8)(9,12,13,21)(10,16,14,19)(11,17,18,20)(15,23,22,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,17)(13,18)(14,22)(16,23)(19,24)(20,21)(25,27)(26,30)(28,33)(29,34)(31,37)(32,38)(35,39)(36,40);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,16)(13,14)(17,23)(18,22)(19,21)(20,24)(25,28,29,36)(26,31,32,35)(27,33,34,40)(30,37,38,39)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8667073);