#7988173

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4251 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)(11,18)(16,22)(19,24)(20,23)(25,26)(27,35)(28,38)(29,32)(30,33)(31,34)(36,39)(37,40);(1,2,5,6)(3,7,4,8)(9,10)(11,16)(12,17)(13,21)(14,15)(18,22)(19,23)(20,24)(25,27,32,30)(26,33,29,35)(28,37,31,36)(34,40,38,39);(1,3,5,4)(2,7,6,8)(10,15)(12,21)(16,22)(19,24)(25,28,32,31)(26,34,29,38)(27,36,30,37)(33,39,35,40);(9,11)(10,16)(12,19)(13,20)(14,18)(15,22)(17,23)(21,24)(25,29,32,26)(27,35,30,33)(28,34,31,38)(36,39,37,40);(1,4,5,3)(2,8,6,7)(9,12,14,21)(10,17,15,13)(11,19,18,24)(16,23,22,20)(25,30,32,27)(26,35,29,33)(28,36,31,37)(34,39,38,40);(9,13,14,17)(10,12,15,21)(11,20,18,23)(16,19,22,24)(25,31,32,28)(26,34,29,38)(27,37,30,36)(33,39,35,40)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=h,e^2=ghi,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, 7895369);