#8188146

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	#37725	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4499 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,d^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095342);