#8118697

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4709 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8025893);