#9712853

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

4414 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9620049);