#8348583

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

4639 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:

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

Polycyclic presentation:

2^9|a^2=hi,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^b=egi,f^c=fgi,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, 8255779);