#8196916

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	#42444	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4395 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=g,d^2=gi,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8104112);