#7901266

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

4398 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=i,e^2=gh,f^2=h,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, 7808462);