#8041191

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	#49577	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4204 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7948387);