#8602804

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	#47928	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4610 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)(3,8)(4,6)(5,7)(9,10,13,16)(11,14,18,17)(12,15,21,20)(19,22,24,23)(25,26)(27,32)(28,29)(30,35)(31,33)(34,40)(36,37)(38,39);(2,4)(7,8)(9,10)(11,17)(12,15)(13,16)(14,18)(19,23)(20,21)(22,24)(25,27)(26,32)(28,38)(29,39)(30,33)(31,35)(34,36)(37,40);(1,3,6,5)(2,7,4,8)(11,18)(14,17)(19,24)(22,23)(26,33)(27,35)(29,40)(36,38);(1,2,6,4)(3,7,5,8)(9,10)(11,14)(12,20)(13,16)(15,21)(17,18)(19,23)(22,24)(25,28,31,34)(26,29,33,40)(27,36,35,38)(30,39,32,37);(1,4,6,2)(3,7,5,8)(9,11,13,18)(10,14,16,17)(12,19,21,24)(15,22,20,23)(25,29)(26,34)(27,37)(28,33)(30,36)(31,40)(32,38)(35,39);(1,5,6,3)(2,7,4,8)(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(25,30)(26,35)(27,33)(28,37)(29,36)(31,32)(34,39)(38,40)

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8510000);