#8850619

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

4480 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)(3,5)(4,8)(6,7)(9,10,14,15)(11,18,16,12)(13,21,22,17)(19,20,23,24)(25,26)(27,30)(28,36)(29,33)(31,35)(32,38)(34,40)(37,39);(2,7)(3,8)(10,15)(12,18)(13,22)(19,23)(25,27,29,35)(26,30,33,31)(28,37,38,34)(32,40,36,39);(1,3,6,8)(2,4,7,5)(10,15)(11,16)(17,21)(19,23)(25,26,29,33)(27,31,35,30)(28,32,38,36)(34,40,37,39);(1,4,6,5)(2,3,7,8)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,24)(22,23)(25,26)(27,30)(28,32)(29,33)(31,35)(34,39)(36,38)(37,40);(1,2)(3,4)(5,8)(6,7)(9,12)(10,16)(11,15)(13,20)(14,18)(17,23)(19,21)(22,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,40)(36,39)(37,38);(1,5,6,4)(2,8,7,3)(9,13,14,22)(10,17,15,21)(11,19,16,23)(12,20,18,24)(25,28)(26,32)(27,34)(29,38)(30,39)(31,40)(33,36)(35,37)

Polycyclic presentation:

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

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, 8757815);