#8509516

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

4336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8416712);