#7879867

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	#48784	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4359 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7787063);