#8687566

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	#44889	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4365 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,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, 8594762);