#8888101

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

4473 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)(3,5,7,8)(11,17)(14,18)(19,23)(22,24)(25,26)(27,32)(28,36)(29,39)(30,31)(33,38)(34,37)(35,40);(3,7)(5,8)(9,10)(11,14)(12,20)(13,16)(15,21)(17,18)(19,24)(22,23)(26,31)(32,40)(33,36)(34,39);(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,14,17,18)(12,15,21,20)(19,22,23,24)(25,27,30,35)(26,32,31,40)(28,37,38,29)(33,39,36,34);(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,21)(22,23)(25,28,30,38)(26,33,31,36)(27,29,35,37)(32,34,40,39);(1,3,4,7)(2,5,6,8)(9,11,13,17)(10,14,16,18)(12,19,21,23)(15,22,20,24)(25,29)(26,34)(27,28)(30,37)(31,39)(32,33)(35,38)(36,40);(1,2)(3,5)(4,6)(7,8)(9,12,13,21)(10,15,16,20)(11,19,17,23)(14,22,18,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,40)(36,38)(37,39)

Polycyclic presentation:

2^9|a^2=h,c^2=gi,d^2=g,e^2=hi,f^2=i,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, 8795297);