#7891989

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

4514 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)(11,16)(12,15)(13,21)(14,18)(17,23)(19,20)(22,24)(25,26)(27,34)(28,36)(29,32)(30,33)(31,38)(35,37)(39,40);(1,2,5,6)(3,7,4,8)(9,11)(10,16)(12,20)(13,18)(14,21)(15,19)(17,24)(22,23)(25,27,30,32)(26,29,33,34)(28,35,38,39)(31,37,36,40);(1,3,5,4)(2,7,6,8)(10,12)(11,19)(17,21)(18,22)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,38)(37,39);(9,12,15,10)(11,20,19,16)(13,21,23,17)(14,22,24,18)(25,28)(26,31)(27,35)(29,37)(30,38)(32,39)(33,36)(34,40);(9,13)(10,17)(11,18)(12,21)(14,20)(15,23)(16,24)(19,22)(25,26)(27,29)(28,31)(30,33)(32,34)(35,37)(36,38)(39,40);(1,4,5,3)(2,8,6,7)(9,14)(10,18)(11,21)(12,22)(13,16)(15,24)(17,19)(20,23)(25,29)(26,32)(27,33)(28,37)(30,34)(31,39)(35,36)(38,40)

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,d^2=h,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, 7799185);