#7944519

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4282 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)(9,10,14,13)(11,17,19,15)(12,20,21,16)(18,22,24,23)(28,36)(33,38)(35,37)(39,40);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,26,31,30)(27,34,29,32)(28,33,36,38)(35,40,37,39);(9,11,14,19)(10,15,13,17)(12,18,21,24)(16,22,20,23)(25,27,31,29)(26,32,30,34)(28,35,36,37)(33,39,38,40);(1,3,5,4)(2,7,6,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,28,31,36)(26,33,30,38)(27,35,29,37)(32,39,34,40);(1,4,5,3)(2,8,6,7)(9,12,14,21)(10,16,13,20)(11,18,19,24)(15,22,17,23)(25,29,31,27)(26,32,30,34)(28,37,36,35)(33,39,38,40);(9,13,14,10)(11,17,19,15)(12,16,21,20)(18,22,24,23)(25,30,31,26)(27,34,29,32)(28,38,36,33)(35,40,37,39)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=i,d^2=gi,e^2=ghi,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, 7851715);