#7917226

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

4458 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)(11,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,24)(25,26,31,33)(27,37,38,32)(28,30,36,34)(29,40,39,35);(1,2)(3,7)(4,8)(5,6)(25,27)(26,32)(28,39)(29,36)(30,40)(31,38)(33,37)(34,35);(1,3,5,4)(2,7,6,8)(10,15)(11,16)(17,22)(18,23)(26,33)(27,38)(28,36)(35,40);(1,3,5,4)(2,7,6,8)(9,11,14,16)(10,12,15,20)(13,18,21,23)(17,19,22,24)(25,28,31,36)(26,34,33,30)(27,39,38,29)(32,35,37,40);(1,3,5,4)(2,7,6,8)(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,29,31,39)(26,35,33,40)(27,36,38,28)(30,32,34,37);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,19,20,24)(25,30,31,34)(26,36,33,28)(27,40,38,35)(29,37,39,32)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7824422);