#8057914

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

4191 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=h,e^2=i,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, 7965110);