#8760684

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	#51357	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4202 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=h,e^2=g,f^2=gh,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, 8667880);