#7977546

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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=ghi,d^2=g,e^2=g,f^2=gh,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, 7884742);