#8747930

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	#36696	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4195 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=g,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8655126);