#8395255

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4512 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)(11,19)(12,20)(13,18)(14,16)(15,21)(17,23)(22,24)(25,26,32,36)(27,28,34,30)(29,33,37,39)(31,35,40,38);(1,2)(3,8)(4,7)(5,6)(9,11,14,21)(10,15,16,19)(12,18,23,22)(13,20,24,17)(25,27)(26,28)(29,31)(30,36)(32,34)(33,35)(37,40)(38,39);(2,6)(4,8)(10,16)(13,24)(15,19)(17,20)(25,28,32,30)(26,27,36,34)(29,35,37,38)(31,39,40,33);(2,6)(3,7)(9,12)(10,17)(11,18)(13,15)(14,23)(16,20)(19,24)(21,22)(25,29)(26,33)(27,31)(28,35)(30,38)(32,37)(34,40)(36,39);(1,3,5,7)(2,4,6,8)(9,11)(10,15)(12,22)(13,20)(14,21)(16,19)(17,24)(18,23)(25,30,32,28)(26,34,36,27)(29,35,37,38)(31,39,40,33);(1,4)(2,7)(3,6)(5,8)(9,13,14,24)(10,18,16,22)(11,20,21,17)(12,15,23,19)(25,31)(26,35)(27,29)(28,33)(30,39)(32,40)(34,37)(36,38)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8302451);