#7981207

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

4544 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)(9,10,14,19)(11,15,21,16)(12,22,23,17)(13,18,24,20)(25,26,30,31)(27,28,33,36)(29,32,38,39)(34,35,37,40);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,20)(13,22)(14,21)(16,19)(17,24)(18,23)(26,31)(27,33)(32,39)(34,37);(9,11,14,21)(10,16,19,15)(12,18,23,20)(13,17,24,22)(25,27)(26,28)(29,37)(30,33)(31,36)(32,40)(34,38)(35,39);(1,2,5,4)(3,6,8,7)(9,11)(10,16)(12,20)(13,17)(14,21)(15,19)(18,23)(22,24)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,39)(38,40);(1,4,5,2)(3,7,8,6)(9,12)(10,17)(11,20)(13,16)(14,23)(15,24)(18,21)(19,22)(25,29,30,38)(26,32,31,39)(27,34,33,37)(28,35,36,40);(9,13)(10,18)(11,17)(12,15)(14,24)(16,23)(19,20)(21,22)(25,27)(26,28)(29,37)(30,33)(31,36)(32,40)(34,38)(35,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=ghi,e^2=g,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, 7888403);