#8199218

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

4278 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,d^2=gh,e^2=gh,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, 8106414);