#8097737

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

4573 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)(3,7)(4,5)(6,8)(9,10,15,14)(11,18,13,16)(12,17,20,22)(19,24,21,23)(25,26,31,36)(27,30,35,32)(28,33,38,40)(29,39,37,34);(2,5)(6,7)(12,20)(17,22)(19,21)(23,24)(25,27)(26,32)(28,37)(29,38)(30,36)(31,35)(33,39)(34,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,13)(10,16,14,18)(12,19,20,21)(17,23,22,24)(25,28,31,38)(26,33,36,40)(27,37,35,29)(30,34,32,39);(1,3,4,8)(2,6,5,7)(9,12,15,20)(10,17,14,22)(11,19,13,21)(16,23,18,24)(25,29,31,37)(26,34,36,39)(27,38,35,28)(30,33,32,40);(9,13,15,11)(10,16,14,18)(12,21,20,19)(17,23,22,24)(25,27)(26,30)(28,37)(29,38)(31,35)(32,36)(33,34)(39,40);(9,14,15,10)(11,18,13,16)(12,22,20,17)(19,24,21,23)(25,30)(26,35)(27,36)(28,34)(29,33)(31,32)(37,40)(38,39)

Polycyclic presentation:

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

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, 8004933);