#9912954

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

4695 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)(4,6)(5,8)(9,10,15,17)(11,18,12,13)(14,16,23,24)(19,22,20,21)(25,26)(27,32)(28,38)(29,34)(30,31)(33,39)(35,40)(36,37);(1,2,4,6)(3,5,8,7)(9,11,15,12)(10,13,17,18)(14,19,23,20)(16,21,24,22)(25,27,31,37)(26,32,30,36)(28,35,39,29)(33,34,38,40);(1,2)(3,5)(4,6)(7,8)(25,28,31,39)(26,33,30,38)(27,29,37,35)(32,40,36,34);(1,3,4,8)(2,5,6,7)(9,12,15,11)(10,13,17,18)(14,20,23,19)(16,21,24,22)(25,28)(26,33)(27,35)(29,37)(30,38)(31,39)(32,34)(36,40);(9,13)(10,11)(12,17)(14,22)(15,18)(16,20)(19,24)(21,23)(25,29,31,35)(26,34,30,40)(27,28,37,39)(32,38,36,33);(9,14)(10,16)(11,19)(12,20)(13,21)(15,23)(17,24)(18,22)(25,30,31,26)(27,36,37,32)(28,38,39,33)(29,34,35,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=g,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, 9820150);