#7812103

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

4382 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7719299);