#7920410

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

4782 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,e^2=gh,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, 7827606);