#8745876

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

4549 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gi,d^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8653072);