#8677171

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	#42598	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4346 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)(11,16)(12,13)(14,18)(15,19)(17,22)(20,23)(21,24)(25,26)(27,36)(28,34)(29,31)(30,33)(32,35)(37,39)(38,40);(1,2)(3,6)(4,7)(5,8)(11,19)(15,16)(20,24)(21,23)(25,27,32,30)(26,33,35,36)(28,37,29,38)(31,39,34,40);(1,3)(2,4)(5,7)(6,8)(9,11)(10,15)(12,20)(13,21)(14,19)(16,18)(17,23)(22,24)(25,28,32,29)(26,34,35,31)(27,37,30,38)(33,40,36,39);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,20)(13,23)(14,19)(15,18)(17,21)(22,24)(25,29,32,28)(26,34,35,31)(27,38,30,37)(33,40,36,39);(1,3)(2,6)(4,8)(5,7)(9,12,14,22)(10,17,18,13)(11,20,19,24)(15,23,16,21)(25,30,32,27)(26,33,35,36)(28,38,29,37)(31,39,34,40);(1,4,5,6)(2,7,8,3)(9,13,14,17)(10,12,18,22)(11,21,19,23)(15,20,16,24)(25,31,32,34)(26,29,35,28)(27,39,30,40)(33,38,36,37)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=g,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, 8584367);