#8944467

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

4501 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8851663);