#8946542

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

4648 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=i,e^2=g,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, 8853738);