#8254768

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

4364 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,e^2=i,f^2=hi,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, 8161964);