#8096356

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4251 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,e^2=g,f^2=g,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, 8003552);