#8227360

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	#36137	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4239 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=ghi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8134556);