#8202088

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

4268 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=h,e^2=g,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, 8109284);