#8395032

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

4411 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=hi,e^2=h,f^2=gh,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, 8302228);