#7732471

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

4417 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7639667);