#8695812

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

4399 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8603008);