#8689775

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

4655 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=h,e^2=h,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, 8596971);