#8745949

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	#52408	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4214 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=hi,d^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8653145);