#8656205

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=hi,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8563401);