#8826322

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

4736 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8733518);