#8132848

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

4549 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=g,e^2=h,f^2=ghi,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, 8040044);