#7983168

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

4472 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)(11,17)(12,14)(13,16)(15,18)(19,24)(20,22)(21,23)(27,33)(31,37)(34,39)(38,40);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,17)(13,21)(14,18)(16,24)(19,22)(20,23)(25,26)(27,31)(28,35)(29,30)(32,36)(33,37)(34,40)(38,39);(1,3,5,4)(2,7,6,8)(10,12)(11,18)(16,20)(19,21)(26,30)(31,37)(32,35)(38,40);(1,3,5,4)(2,7,6,8)(9,11)(10,15)(12,17)(13,19)(14,18)(16,23)(20,24)(21,22)(25,27)(26,31)(28,34)(29,33)(30,37)(32,38)(35,40)(36,39);(1,4,5,3)(2,8,6,7)(9,12,14,10)(11,17,18,15)(13,16,22,20)(19,23,21,24)(25,26)(27,31)(28,35)(29,30)(32,36)(33,37)(34,40)(38,39);(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,24)(18,21)(25,28,29,36)(26,32,30,35)(27,34,33,39)(31,38,37,40)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=gh,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7890364);