#7866444

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

4511 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7773640);