#8132910

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

4268 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:

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

Polycyclic presentation:

2^9|a^2=i,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, 8040106);