#8095765

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

4659 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)(3,7)(9,10)(11,20)(12,22)(13,16)(14,21)(15,17)(18,23)(19,24)(25,26,31,33)(27,32,30,40)(28,29,34,38)(35,39,37,36);(1,2,4,5)(3,6,7,8)(9,11,15,13)(10,16,17,20)(12,19,23,21)(14,18,24,22)(25,27,31,30)(26,32,33,40)(28,37,34,35)(29,36,38,39);(1,3)(2,6)(4,7)(5,8)(10,17)(11,13)(18,22)(19,21)(26,33)(29,38)(32,40)(36,39);(1,2)(3,6)(4,5)(7,8)(9,12,15,23)(10,18,17,22)(11,21,13,19)(14,20,24,16)(25,28,31,34)(26,29,33,38)(27,35,30,37)(32,39,40,36);(9,13,15,11)(10,16,17,20)(12,19,23,21)(14,22,24,18)(25,29)(26,34)(27,36)(28,33)(30,39)(31,38)(32,35)(37,40);(9,14)(10,19)(11,18)(12,20)(13,22)(15,24)(16,23)(17,21)(25,30,31,27)(26,32,33,40)(28,37,34,35)(29,39,38,36)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=ghi,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8002961);