#8678912

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

4563 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8586108);