#8158993

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

4446 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8066189);