#8799056

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

4423 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=gi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8706252);