#8914534

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	#51357	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4315 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)(11,19)(12,15)(13,17)(14,22)(16,20)(18,24)(21,23)(25,26,30,35)(27,31,37,32)(28,33,38,39)(29,36,40,34);(1,2)(3,8)(4,6)(5,7)(9,11,15,20)(10,16,12,19)(13,22,23,18)(14,17,24,21)(25,27)(26,31)(28,34)(29,33)(30,37)(32,35)(36,38)(39,40);(1,3)(2,4)(5,6)(7,8)(10,12)(14,24)(16,19)(17,21)(25,27,30,37)(26,32,35,31)(28,36,38,34)(29,39,40,33);(3,6)(4,8)(9,12,15,10)(11,19,20,16)(13,17,23,21)(14,22,24,18)(25,27)(26,31)(28,34)(29,33)(30,37)(32,35)(36,38)(39,40);(1,4,5,8)(2,6,7,3)(9,13,15,23)(10,17,12,21)(11,18,20,22)(14,19,24,16)(25,28)(26,33)(27,36)(29,32)(30,38)(31,40)(34,37)(35,39);(1,2)(3,8)(4,6)(5,7)(9,14,15,24)(10,18,12,22)(11,17,20,21)(13,19,23,16)(25,29,30,40)(26,34,35,36)(27,33,37,39)(28,32,38,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=g,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8821730);