#8678868

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	#45046	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4364 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)(11,21)(12,18)(13,15)(14,19)(16,20)(17,23)(22,24)(25,26,30,34)(27,35,38,31)(28,39,37,32)(29,33,36,40);(1,2)(3,8)(4,6)(5,7)(9,11)(10,17)(12,14)(13,22)(15,19)(16,23)(18,24)(20,21)(25,27)(26,31)(28,36)(29,37)(30,38)(32,40)(33,39)(34,35);(1,3,5,4)(2,6,7,8)(9,12,16,13)(10,18,20,15)(11,14,23,22)(17,24,21,19)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,40)(37,39);(1,2)(3,6)(4,8)(5,7)(9,13,16,12)(10,18,20,15)(11,22,23,14)(17,24,21,19)(25,26)(27,35)(28,32)(29,40)(30,34)(31,38)(33,36)(37,39);(1,2)(3,6)(4,8)(5,7)(9,14,16,22)(10,19,20,24)(11,12,23,13)(15,21,18,17)(25,28)(26,32)(27,36)(29,38)(30,37)(31,40)(33,35)(34,39);(1,4,5,3)(2,6,7,8)(9,15,16,18)(10,13,20,12)(11,19,23,24)(14,17,22,21)(25,29,30,36)(26,33,34,40)(27,37,38,28)(31,39,35,32)

Polycyclic presentation:

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

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, 8586064);