#7645836

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	#39118	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4287 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)(12,15)(13,21)(14,17)(16,22)(19,23)(20,24)(26,30)(27,33)(32,38)(34,35);(1,2)(3,7)(4,8)(5,6)(9,11,14,12)(10,15,17,18)(13,19,22,20)(16,23,21,24);(1,3,5,4)(2,7,6,8)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,23)(21,22)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,40)(36,38);(1,3,5,4)(2,7,6,8)(9,12,14,11)(10,15,17,18)(13,20,22,19)(16,23,21,24)(25,27,29,33)(26,31,30,37)(28,35,36,34)(32,40,38,39);(9,10)(11,15)(12,18)(13,16)(14,17)(19,23)(20,24)(21,22)(25,28,29,36)(26,32,30,38)(27,34,33,35)(31,39,37,40);(1,4,5,3)(2,8,6,7)(9,13,14,22)(10,16,17,21)(11,19,12,20)(15,23,18,24)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,40)(36,38)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7553032);