#8137620

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

4701 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=gh,e^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, 8044816);