#8132938

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
prime radical	#34741	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4337 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,e^2=g,f^2=gi,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, 8040134);