#8097792

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

4617 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)(9,10)(11,16)(12,20)(13,17)(14,19)(15,21)(18,22)(23,24)(25,26,31,34)(27,35,29,32)(28,30,33,40)(36,39,38,37);(2,5)(3,7)(12,21)(15,20)(18,24)(22,23)(25,27,31,29)(26,32,34,35)(28,36,33,38)(30,37,40,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,28,31,33)(26,30,34,40)(27,36,29,38)(32,37,35,39);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,18)(13,21)(14,22)(16,23)(17,20)(19,24);(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,22)(21,24)(25,29,31,27)(26,32,34,35)(28,38,33,36)(30,37,40,39);(9,10)(11,14)(12,15)(13,17)(16,19)(18,22)(20,21)(23,24)(25,30)(26,33)(27,37)(28,34)(29,39)(31,40)(32,38)(35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dhi,c^b=ci,d^b=dh,f^e=fghi

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, 8004988);