#8274039

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	#52311	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4431 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8181235);