#7868923

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

4496 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)(9,10,15,16)(11,14,19,24)(12,17,21,23)(13,22,20,18);(1,2,5,6)(3,7,4,8)(9,11,15,19)(10,14,16,24)(12,20,21,13)(17,18,23,22)(25,26)(27,32)(28,31)(29,33)(30,35)(34,38)(36,39)(37,40);(1,3,5,4)(2,7,6,8)(10,16)(11,19)(12,21)(18,22)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(9,12)(10,17)(11,20)(13,19)(14,18)(15,21)(16,23)(22,24)(25,28,29,36)(26,31,33,39)(27,34,35,40)(30,37,32,38);(1,3,5,4)(2,7,6,8)(9,13,15,20)(10,18,16,22)(11,12,19,21)(14,23,24,17)(25,27)(26,32)(28,34)(29,35)(30,33)(31,38)(36,40)(37,39);(1,4,5,3)(2,8,6,7)(9,14)(10,19)(11,16)(12,18)(13,17)(15,24)(20,23)(21,22)(25,26)(27,30)(28,31)(29,33)(32,35)(34,37)(36,39)(38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=hi,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, 7776119);