#8340188

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	#52280	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4298 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)(12,17)(14,19)(15,16)(18,23)(20,22)(21,24)(25,26,31,36)(27,34,29,32)(28,37,39,33)(30,35,40,38);(3,7)(6,8)(9,11)(10,13)(12,14)(15,18)(16,23)(17,19)(20,21)(22,24)(25,27,31,29)(26,32,36,34)(28,38,39,35)(30,37,40,33);(1,2)(3,6)(4,5)(7,8)(10,16)(11,18)(14,20)(17,24)(25,27,31,29)(26,32,36,34)(28,35,39,38)(30,33,40,37);(1,3)(2,6)(4,7)(5,8)(9,12)(10,17)(11,20)(13,22)(14,18)(15,21)(16,24)(19,23)(25,28)(26,33)(27,35)(29,38)(30,34)(31,39)(32,40)(36,37);(1,2)(3,6)(4,5)(7,8)(9,13,15,23)(10,18,16,11)(12,19,21,22)(14,17,20,24)(25,29,31,27)(26,34,36,32)(28,35,39,38)(30,33,40,37);(9,14,15,20)(10,19,16,22)(11,12,18,21)(13,17,23,24)(25,30,31,40)(26,35,36,38)(27,37,29,33)(28,34,39,32)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8247384);