#8107345

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

4473 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:

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

Polycyclic presentation:

2^9|d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

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, 8014541);