#8730677

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

4571 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8637873);