#7979800

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
prime radical	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4416 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7886996);