#8935338

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

4697 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=gh,f^2=g,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, 8842534);