#9923983

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

4227 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9831179);