#8271906

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

4651 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)(3,7)(4,5)(6,8)(9,10,15,16)(11,17,12,21)(13,18,24,14)(19,20,22,23)(26,30)(27,33)(32,37)(34,39);(1,3)(2,6)(4,7)(5,8)(10,16)(13,24)(17,21)(19,22)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,2,5,4)(3,6,8,7)(9,11,15,12)(10,17,16,21)(13,19,24,22)(14,23,18,20)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,2,5,4)(3,7,8,6)(9,12,15,11)(10,17,16,21)(13,22,24,19)(14,23,18,20)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,4,5,2)(3,6,8,7)(9,13)(10,18)(11,19)(12,22)(14,16)(15,24)(17,20)(21,23)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(1,3)(2,7)(4,6)(5,8)(9,14,15,18)(10,13,16,24)(11,20,12,23)(17,22,21,19)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)

Polycyclic presentation:

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

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, 8179102);