#7987131

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4317 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)(9,10,14,18)(11,12,16,15)(13,17,23,24)(19,20,21,22)(25,26)(27,36)(28,30)(29,40)(31,33)(32,37)(34,39)(35,38);(1,2,5,6)(3,7,4,8)(9,11,14,16)(10,15,18,12)(13,21,23,19)(17,20,24,22)(25,27,31,37)(26,32,33,36)(28,35,39,40)(29,34,38,30);(1,3,5,4)(2,7,6,8)(11,16)(12,15)(19,21)(20,22)(26,33)(27,37)(28,39)(29,38);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,21)(23,24)(25,28)(26,34)(27,35)(29,36)(30,33)(31,39)(32,38)(37,40);(1,4,5,3)(2,8,6,7)(9,12,14,15)(10,16,18,11)(13,22,23,20)(17,19,24,21)(25,29)(26,35)(27,34)(28,36)(30,37)(31,38)(32,39)(33,40);(9,13,14,23)(10,17,18,24)(11,19,16,21)(12,20,15,22)(25,30,31,34)(26,28,33,39)(27,38,37,29)(32,40,36,35)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gi,d^2=gi,e^2=gh,f^2=i,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, 7894327);