#8141425

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

4461 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8048621);