#8118782

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

4695 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)(3,4)(9,10)(11,17)(12,14)(13,20)(15,19)(16,21)(18,22)(23,24)(25,26)(27,28)(29,37)(30,31)(32,34)(33,38)(35,39)(36,40);(1,2,6,5)(3,8,4,7)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,39)(38,40);(1,3,6,4)(2,7,5,8)(10,12)(15,17)(16,20)(22,23)(26,31)(27,34)(33,37)(35,40);(1,4,6,3)(2,8,5,7)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,28,30,32)(26,27,31,34)(29,36,38,39)(33,35,37,40);(1,5,6,2)(3,8,4,7)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,27)(26,32)(28,31)(29,35)(30,34)(33,39)(36,37)(38,40);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,37)(32,39)(34,40)

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8025978);