#7919320

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	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4200 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)(11,18)(12,22)(13,19)(14,17)(15,16)(20,21)(23,24)(25,26,32,31)(27,28,34,30)(29,33,39,38)(35,36,40,37);(1,2)(3,7)(4,8)(5,6)(9,11)(10,13)(12,20)(14,21)(15,19)(16,18)(17,23)(22,24)(25,27)(26,30)(28,31)(29,35)(32,34)(33,37)(36,38)(39,40);(1,3,5,4)(2,7,6,8)(10,16)(11,19)(12,17)(21,24)(25,28,32,30)(26,27,31,34)(29,37,39,36)(33,40,38,35);(1,3,5,4)(2,7,6,8)(9,12,15,17)(10,14,16,22)(11,20,19,23)(13,21,18,24)(25,29,32,39)(26,33,31,38)(27,35,34,40)(28,36,30,37);(9,13)(10,11)(12,21)(14,20)(15,18)(16,19)(17,24)(22,23)(25,30,32,28)(26,34,31,27)(29,37,39,36)(33,40,38,35);(1,4,5,3)(2,8,6,7)(9,14)(10,17)(11,21)(12,16)(13,23)(15,22)(18,20)(19,24)(25,31,32,26)(27,28,34,30)(29,38,39,33)(35,36,40,37)

Polycyclic presentation:

2^9|a^2=hi,c^2=gi,d^2=g,e^2=hi,f^2=gi,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, 7826516);