#8057906

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

4315 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)(11,16,19,12)(13,21,22,17)(18,20,24,23)(25,26,29,32)(27,33,35,30)(28,31,38,37)(34,40,36,39);(1,2,5,6)(3,7,4,8)(10,15)(12,16)(17,21)(20,23)(25,27)(26,30)(28,36)(29,35)(31,40)(32,33)(34,38)(37,39);(9,11,14,19)(10,12,15,16)(13,18,22,24)(17,20,21,23)(25,27,29,35)(26,30,32,33)(28,34,38,36)(31,39,37,40);(1,3,5,4)(2,7,6,8)(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,21)(22,24)(25,27)(26,30)(28,34)(29,35)(31,39)(32,33)(36,38)(37,40);(1,4,5,3)(2,8,6,7)(9,12,14,16)(10,11,15,19)(13,20,22,23)(17,18,21,24)(25,28)(26,31)(27,34)(29,38)(30,39)(32,37)(33,40)(35,36);(9,13,14,22)(10,17,15,21)(11,18,19,24)(12,20,16,23)(25,26)(27,30)(28,37)(29,32)(31,38)(33,35)(34,40)(36,39)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=gh,e^2=gi,f^2=hi,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, 7965102);