#8678964

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	#2541	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4375 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)(3,8)(4,5)(6,7)(10,15)(11,20)(12,19)(22,23)(25,26,32,29)(27,33,31,35)(28,30,39,34)(36,40,38,37);(1,3,6,5)(2,4,7,8)(9,10)(11,16)(12,18)(13,19)(14,15)(17,20)(21,23)(22,24)(25,27,32,31)(26,33,29,35)(28,38,39,36)(30,37,34,40);(9,11)(10,16)(12,21)(13,22)(14,20)(15,17)(18,23)(19,24)(25,28,32,39)(26,34,29,30)(27,36,31,38)(33,37,35,40);(1,4,6,8)(2,3,7,5)(9,11)(10,17)(12,21)(13,23)(14,20)(15,16)(18,22)(19,24)(25,29,32,26)(27,33,31,35)(28,30,39,34)(36,37,38,40);(1,2)(3,4)(5,8)(6,7)(9,12)(10,18)(11,21)(13,15)(14,19)(16,23)(17,22)(20,24)(25,30)(26,28)(27,37)(29,39)(31,40)(32,34)(33,38)(35,36);(1,5,6,3)(2,4,7,8)(9,13,14,18)(10,19,15,12)(11,22,20,23)(16,24,17,21)(25,31,32,27)(26,35,29,33)(28,38,39,36)(30,37,34,40)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^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, 8586160);