#8291687

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

4521 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)(6,8)(9,10)(11,19)(12,21)(13,18)(14,15)(16,20)(17,23)(22,24)(25,26,30,28)(27,31,33,34)(29,32,37,36)(35,38,39,40);(3,7)(6,8)(10,15)(11,20)(17,21)(18,22)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,2)(3,6)(4,5)(7,8)(10,15)(13,24)(16,19)(17,21)(27,33)(31,34)(35,39)(38,40);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,22)(13,21)(14,20)(15,19)(17,24)(18,23)(25,28,30,26)(27,34,33,31)(29,32,37,36)(35,38,39,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,23)(10,17,15,21)(11,18,20,22)(13,19,24,16)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40);(9,13)(10,18)(11,21)(12,19)(14,24)(15,22)(16,23)(17,20)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39)

Polycyclic presentation:

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

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, 8198883);