#7806843

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

4366 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,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, 7714039);