#8165566

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

4625 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=g,e^2=gi,f^2=h,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^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, 8072762);