#8139524

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	#43887	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4701 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:

(2,5)(6,8)(11,18)(13,21)(15,16)(17,20)(25,26,30,29)(27,33,35,31)(28,36,37,32)(34,38,40,39);(3,7)(6,8)(9,10)(11,16)(12,14)(13,17)(15,18)(19,23)(20,21)(22,24)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,16)(13,19)(14,18)(17,22)(20,23)(21,24)(25,26,30,29)(27,31,35,33)(28,32,37,36)(34,38,40,39);(1,3)(2,6)(4,7)(5,8)(9,12,14,10)(11,16,18,15)(13,20,21,17)(19,23,24,22)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,15)(13,19)(14,18)(17,23)(20,22)(21,24)(25,29,30,26)(27,33,35,31)(28,32,37,36)(34,38,40,39);(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(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=g,d^2=i,e^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, 8046720);