#8125659

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	#42023	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4358 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=gh,e^2=g,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^c=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, 8032855);