#8677440

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	#52297	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4343 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,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, 8584636);