#8751136

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

4329 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=gi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

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, 8658332);