#8445471

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

4599 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=gh,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8352667);