#8359161

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

4549 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=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, 8266357);