#8687375

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

4480 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)(6,8)(10,14)(11,17)(16,19)(18,20)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,38)(39,40);(3,7)(6,8)(10,14)(15,22)(16,19)(23,24)(25,27,31,38)(26,32,36,37)(28,29,35,39)(30,34,33,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,28,31,35)(26,33,36,30)(27,29,38,39)(32,40,37,34);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,19)(13,14)(16,21)(17,22)(18,24)(20,23)(25,29)(26,34)(27,35)(28,38)(30,37)(31,39)(32,33)(36,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,17)(14,22)(16,24)(18,21)(19,23)(25,29)(26,34)(27,35)(28,38)(30,37)(31,39)(32,33)(36,40);(9,12,13,21)(10,16,14,19)(11,18,17,20)(15,23,22,24)(25,30,31,33)(26,35,36,28)(27,34,38,40)(29,32,39,37)

Polycyclic presentation:

2^9|b^2=g,c^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,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, 8594571);