#8096279

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	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4273 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)(9,10,15,19)(11,17,12,16)(13,14,18,24)(20,23,22,21)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,38)(37,39);(1,3,7,6)(2,5,8,4)(9,11,15,12)(10,16,19,17)(13,20,18,22)(14,21,24,23)(28,36)(31,38)(35,39)(37,40);(1,4,7,5)(2,3,8,6)(25,27,30,32)(26,29,33,34)(28,35,36,39)(31,37,38,40);(1,4,7,5)(2,3,8,6)(9,12,15,11)(10,17,19,16)(13,22,18,20)(14,23,24,21)(25,28)(26,31)(27,35)(29,37)(30,36)(32,39)(33,38)(34,40);(1,5,7,4)(2,3,8,6)(9,13)(10,14)(11,20)(12,22)(15,18)(16,21)(17,23)(19,24)(25,26)(27,29)(28,31)(30,33)(32,34)(35,37)(36,38)(39,40);(1,6,7,3)(2,5,8,4)(9,14)(10,18)(11,21)(12,23)(13,19)(15,24)(16,22)(17,20)(25,29)(26,32)(27,33)(28,37)(30,34)(31,39)(35,38)(36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=g,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, 8003475);