#8323148

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

4461 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)(3,7)(9,10,14,17)(11,12,19,15)(13,16,21,23)(18,20,24,22)(25,26)(27,35)(28,30)(29,32)(31,33)(34,39)(36,37)(38,40);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,27,31,29)(26,32,33,35)(28,38,39,36)(30,37,34,40);(1,2)(3,6)(4,5)(7,8)(13,21)(16,23)(18,24)(20,22)(26,33)(30,34)(32,35)(37,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,15)(10,11,17,19)(13,20,21,22)(16,18,23,24)(25,28)(26,34)(27,36)(29,38)(30,33)(31,39)(32,37)(35,40);(1,3,4,7)(2,6,5,8)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,29,31,27)(26,35,33,32)(28,36,39,38)(30,40,34,37);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,34)(26,28,33,39)(27,37,29,40)(32,38,35,36)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=gh,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8230344);