#8841750

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

4482 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8748946);