#8114363

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

4400 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8021559);