#7974997

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

4567 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)(9,10,14,16)(11,15,20,17)(12,13,19,18)(21,23,24,22)(25,26,31,28)(27,33,29,32)(30,34,38,36)(35,40,37,39);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,21)(13,23)(14,20)(16,17)(18,22)(19,24)(25,27,31,29)(26,32,28,33)(30,35,38,37)(34,39,36,40);(1,2,5,4)(3,6,8,7)(10,16)(11,20)(12,19)(22,23)(26,28)(32,33)(34,36)(39,40);(1,2,5,4)(3,6,8,7)(9,11)(10,17)(12,21)(13,22)(14,20)(15,16)(18,23)(19,24)(25,28,31,26)(27,33,29,32)(30,36,38,34)(35,40,37,39);(1,4,5,2)(3,7,8,6)(9,12)(10,18)(11,21)(13,16)(14,19)(15,22)(17,23)(20,24)(25,29,31,27)(26,33,28,32)(30,37,38,35)(34,40,36,39);(9,13,14,18)(10,19,16,12)(11,22,20,23)(15,21,17,24)(25,30,31,38)(26,34,28,36)(27,35,29,37)(32,39,33,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=ghi,d^2=g,e^2=g,f^2=h,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, 7882193);