#8687572

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

4473 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=h,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, 8594768);