#8258576

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	#45451	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4398 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)(11,16)(12,18)(13,20)(14,19)(15,17)(21,24)(22,23)(25,26,30,29)(27,33,35,31)(28,36,37,32)(34,38,40,39);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,14)(13,17)(15,20)(18,19)(21,22)(23,24)(25,27,30,35)(26,31,29,33)(28,34,37,40)(32,38,36,39);(1,2,4,6)(3,5,8,7)(10,17)(11,20)(14,21)(18,23)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,18)(11,21)(13,19)(14,20)(15,22)(16,24)(17,23)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(1,3)(2,5)(4,8)(6,7)(9,13)(10,11)(12,19)(14,23)(15,16)(17,20)(18,21)(22,24)(25,28,30,37)(26,32,29,36)(27,34,35,40)(31,38,33,39);(1,2)(3,5)(4,6)(7,8)(9,14,15,21)(10,19,17,24)(11,12,20,22)(13,23,16,18)(25,29,30,26)(27,33,35,31)(28,36,37,32)(34,39,40,38)

Polycyclic presentation:

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

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, 8165772);