#7732819

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

4629 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,d^2=g,e^2=hi,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, 7640015);