#8389368

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	#57189	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4566 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=g,d^2=gh,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, 8296564);