#8665278

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

4397 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:

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

Polycyclic presentation:

2^9|c^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8572474);