#8258610

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4271 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)(3,7)(4,5)(6,8)(9,10)(11,18)(12,13)(14,15)(16,19)(17,22)(20,23)(21,24)(25,26,30,29)(27,31,33,37)(28,32,35,36)(34,38,39,40);(2,5)(6,7)(10,15)(11,19)(12,22)(21,23)(27,33)(28,35)(31,37)(32,36);(1,2)(3,6)(4,5)(7,8)(10,15)(13,17)(16,18)(21,23)(25,27,30,33)(26,31,29,37)(28,34,35,39)(32,38,36,40);(1,3,4,8)(2,6,5,7)(9,11,14,19)(10,16,15,18)(12,20,22,24)(13,21,17,23);(9,12)(10,17)(11,20)(13,15)(14,22)(16,23)(18,21)(19,24)(25,28,30,35)(26,32,29,36)(27,34,33,39)(31,38,37,40);(9,13,14,17)(10,12,15,22)(11,21,19,23)(16,20,18,24)(25,29,30,26)(27,31,33,37)(28,36,35,32)(34,38,39,40)

Polycyclic presentation:

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

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, 8165806);