#7891241

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

4555 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)(21,24)(22,23);(1,2)(3,7)(4,8)(5,6)(9,10,14,15)(11,12,16,20)(13,17,22,23)(18,19,24,21)(25,26,31,30)(27,35,38,32)(28,33,37,39)(29,34,36,40);(1,3,5,4)(2,7,6,8)(10,15)(11,16)(17,23)(18,24)(25,27)(26,32)(28,36)(29,37)(30,35)(31,38)(33,34)(39,40);(1,3,5,4)(2,7,6,8)(9,11,14,16)(10,12,15,20)(13,18,22,24)(17,19,23,21)(25,28,31,37)(26,33,30,39)(27,36,38,29)(32,34,35,40);(1,3,5,4)(2,7,6,8)(9,12)(10,16)(11,15)(13,21)(14,20)(17,18)(19,22)(23,24)(25,29,31,36)(26,34,30,40)(27,37,38,28)(32,33,35,39);(1,4,5,3)(2,8,6,7)(9,13,14,22)(10,17,15,23)(11,18,16,24)(12,19,20,21)(25,30,31,26)(27,35,38,32)(28,39,37,33)(29,34,36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7798437);