#7868926

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

4468 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=ghi,e^2=gh,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, 7776122);