#7952523

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

4635 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:

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

Polycyclic presentation:

2^9|b^2=ghi,d^2=i,e^2=h,f^2=g,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, 7859719);