#7927409

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	#44938	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4367 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=g,d^2=h,e^2=gh,f^2=i,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, 7834605);