#8704183

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

4667 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)(11,12)(13,17)(14,15)(16,18)(19,20)(21,22)(23,24)(25,26,31,36)(27,37,39,32)(28,30,35,33)(29,40,38,34);(1,2)(3,6)(4,7)(5,8)(9,10)(11,18)(12,16)(13,17)(14,15)(19,22)(20,21)(23,24)(25,27)(26,32)(28,29)(30,34)(31,39)(33,40)(35,38)(36,37);(1,3)(2,5)(4,8)(6,7)(10,15)(11,16)(17,24)(19,21)(25,28,31,35)(26,33,36,30)(27,29,39,38)(32,40,37,34);(3,8)(5,6)(9,11,14,16)(10,12,15,18)(13,21,23,19)(17,22,24,20)(25,28)(26,33)(27,38)(29,39)(30,36)(31,35)(32,34)(37,40);(1,2)(3,5)(4,7)(6,8)(9,12)(10,16)(11,15)(13,22)(14,18)(17,19)(20,23)(21,24)(25,29,31,38)(26,34,36,40)(27,28,39,35)(30,37,33,32);(1,3)(2,6)(4,8)(5,7)(9,13,14,23)(10,17,15,24)(11,19,16,21)(12,20,18,22)(25,30,31,33)(26,35,36,28)(27,34,39,40)(29,32,38,37)

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8611379);