#7983185

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

4293 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gh,f^2=hi,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, 7890381);