#8057942

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

4331 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=gh,e^2=gi,f^2=ghi,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, 7965138);