#8214772

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

4407 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:

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

Polycyclic presentation:

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

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, 8121968);