#8096551

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	#2541	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4345 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)(3,4,6,5)(9,10,14,18)(11,13,17,15)(12,16,20,23)(19,21,24,22)(26,32)(28,33)(34,40)(35,38);(1,3,7,6)(2,5,8,4)(12,20)(16,23)(19,24)(21,22)(25,26,31,32)(27,33,37,28)(29,34,36,40)(30,35,39,38);(1,4,7,5)(2,3,8,6)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,27,31,37)(26,33,32,28)(29,30,36,39)(34,35,40,38);(1,4,7,5)(2,3,8,6)(9,12,14,20)(10,16,18,23)(11,19,17,24)(13,21,15,22)(25,28)(26,27)(29,38)(30,34)(31,33)(32,37)(35,36)(39,40);(1,5,7,4)(2,3,8,6)(9,11)(10,13)(12,19)(14,17)(15,18)(16,21)(20,24)(22,23)(25,29)(26,34)(27,30)(28,38)(31,36)(32,40)(33,35)(37,39);(1,6,7,3)(2,5,8,4)(9,13)(10,17)(11,18)(12,21)(14,15)(16,24)(19,23)(20,22)(25,30)(26,35)(27,36)(28,34)(29,37)(31,39)(32,38)(33,40)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8003747);