#8761950

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
prime radical	#50634	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4492 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)(3,4)(5,6)(7,8)(9,10,13,14)(11,17,19,15)(12,16,20,22)(18,24,21,23)(25,26,30,33)(27,31,36,32)(28,29,38,40)(34,35,37,39);(2,6)(4,7)(10,14)(12,20)(15,17)(18,21)(25,27)(26,31)(28,37)(29,39)(30,36)(32,33)(34,38)(35,40);(1,2,5,6)(3,7,8,4)(9,11,13,19)(10,15,14,17)(12,21,20,18)(16,24,22,23)(25,27,30,36)(26,32,33,31)(28,37,38,34)(29,35,40,39);(1,3)(2,7)(4,6)(5,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,23)(20,21)(22,24)(25,27)(26,31)(28,34)(29,35)(30,36)(32,33)(37,38)(39,40);(1,4,5,7)(2,3,6,8)(9,12,13,20)(10,16,14,22)(11,18,19,21)(15,23,17,24)(25,28)(26,29)(27,34)(30,38)(31,35)(32,39)(33,40)(36,37);(1,3)(2,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,22)(21,24)(25,29)(26,28)(27,35)(30,40)(31,34)(32,37)(33,38)(36,39)

Polycyclic presentation:

2^9|a^2=gi,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8669146);