#8696276

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

4603 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8603472);