#7892205

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

4502 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)(6,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(26,31)(28,34)(32,33)(35,40);(1,2,4,5)(3,6,7,8)(25,26,30,31)(27,33,37,32)(28,29,34,39)(35,38,40,36);(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23)(25,27)(26,32)(28,35)(29,36)(30,37)(31,33)(34,40)(38,39);(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,28,30,34)(26,29,31,39)(27,35,37,40)(32,36,33,38);(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27)(26,33)(28,35)(29,38)(30,37)(31,32)(34,40)(36,39);(1,3,4,7)(2,6,5,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,29)(26,34)(27,36)(28,31)(30,39)(32,40)(33,35)(37,38)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7799401);