#7806772

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

4519 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7713968);