#8095756

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

4491 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:

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

Polycyclic presentation:

2^9|a^2=g,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, 8002952);