#8258551

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	#36424	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4294 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:

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

Polycyclic presentation:

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

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, 8165747);