#7853363

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

4422 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7760559);