#8758027

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

4570 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8665223);