#8257963

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	#38824	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4181 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gi,d^2=h,e^2=hi,f^2=ghi,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, 8165159);