#8137169

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

4546 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)(10,14)(12,20)(15,17)(18,24)(25,26,30,34)(27,31,32,28)(29,37,38,33)(35,36,40,39);(1,3,6,5)(2,7,8,4)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,27,30,32)(26,31,34,28)(29,35,38,40)(33,39,37,36);(1,4)(2,3)(5,8)(6,7)(25,27)(26,28)(29,35)(30,32)(31,34)(33,36)(37,39)(38,40);(2,8)(3,5)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27)(26,31)(28,34)(29,35)(30,32)(33,39)(36,37)(38,40);(1,5,6,3)(2,4,8,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28,30,31)(26,32,34,27)(29,36,38,39)(33,40,37,35);(1,4)(2,3)(5,8)(6,7)(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,29,30,38)(26,33,34,37)(27,35,32,40)(28,36,31,39)

Polycyclic presentation:

2^9|a^2=h,b^2=h,e^2=h,f^2=gh,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, 8044365);