#8139572

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

4602 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8046768);