#7927642

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

4508 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)(9,10)(11,22)(12,19)(13,24)(14,18)(15,20)(16,21)(17,23)(25,26,29,32)(27,30,35,39)(28,31,36,40)(33,37,34,38);(1,3,5,8)(2,6,4,7)(9,11)(10,17)(12,13)(14,20)(15,19)(16,23)(18,24)(21,22)(25,27)(26,30)(28,34)(29,35)(31,38)(32,39)(33,36)(37,40);(9,12,16,14)(10,18,21,19)(11,20,23,13)(15,22,24,17)(25,28)(26,31)(27,33)(29,36)(30,37)(32,40)(34,35)(38,39);(1,2,5,4)(3,6,8,7)(9,13,16,20)(10,15,21,24)(11,12,23,14)(17,19,22,18)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,39)(36,40);(1,4,5,2)(3,7,8,6)(9,14,16,12)(10,19,21,18)(11,20,23,13)(15,22,24,17)(25,28)(26,31)(27,34)(29,36)(30,38)(32,40)(33,35)(37,39);(9,15)(10,20)(11,19)(12,22)(13,21)(14,17)(16,24)(18,23)(25,27)(26,30)(28,33)(29,35)(31,37)(32,39)(34,36)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7834838);