#7969681

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
prime radical	#49573	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4498 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)(9,10,14,18)(11,19,20,15)(12,16,22,23)(13,17,24,21)(25,26)(27,30)(28,36)(29,33)(31,35)(32,38)(34,40)(37,39);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,21)(13,23)(14,20)(16,24)(17,22)(18,19)(25,27)(26,30)(28,34)(29,35)(31,33)(32,39)(36,40)(37,38);(11,20)(12,22)(15,19)(16,23)(25,27,29,35)(26,31,33,30)(28,37,38,34)(32,39,36,40);(9,10)(11,15)(12,16)(13,17)(14,18)(19,20)(21,24)(22,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,40)(36,39)(37,38);(1,2,5,4)(3,6,8,7)(9,12,14,22)(10,16,18,23)(11,21,20,17)(13,15,24,19)(25,28)(26,32)(27,34)(29,38)(30,39)(31,40)(33,36)(35,37);(1,4,5,2)(3,7,8,6)(9,13,14,24)(10,17,18,21)(11,16,20,23)(12,19,22,15)(25,26)(27,31)(28,36)(29,33)(30,35)(32,38)(34,39)(37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,e^2=g,f^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, 7876877);