#8127285

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

4293 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)(10,15)(11,18)(12,20)(21,23)(29,37)(31,38)(34,40)(36,39);(3,7)(6,8)(9,10,14,15)(11,13,18,16)(12,17,20,22)(19,21,24,23)(25,26,30,32)(27,33,35,28)(29,31,37,38)(34,39,40,36);(1,2)(3,6)(4,5)(7,8)(9,11,14,18)(10,16,15,13)(12,19,20,24)(17,23,22,21)(25,27,30,35)(26,28,32,33)(29,34,37,40)(31,36,38,39);(1,3,4,7)(2,6,5,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,22)(21,24)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,19)(13,21)(14,20)(15,22)(16,23)(18,24)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,39)(38,40);(9,13,14,16)(10,18,15,11)(12,21,20,23)(17,24,22,19)(25,29)(26,31)(27,34)(28,36)(30,37)(32,38)(33,39)(35,40)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=gh,f^2=g,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^c=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, 8034481);