#7969669

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	#45476	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4527 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)(11,17,18,12)(13,15,23,24)(19,22,21,20)(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)(10,12)(13,21)(14,18)(15,22)(16,17)(19,23)(20,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,39)(36,40)(37,38);(11,18)(12,17)(19,21)(20,22)(25,27,29,35)(26,31,33,30)(28,37,38,34)(32,39,36,40);(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,22)(23,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,40)(36,39)(37,38);(9,12)(10,11)(13,22)(14,17)(15,21)(16,18)(19,24)(20,23)(25,28)(26,32)(27,34)(29,38)(30,39)(31,40)(33,36)(35,37);(1,4,5,2)(3,7,8,6)(9,13,14,23)(10,15,16,24)(11,19,18,21)(12,20,17,22)(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,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, 7876865);