#8230706

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

4320 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)(25,26,31,28)(27,32,29,33)(30,34,38,36)(35,39,37,40);(1,3,4,8)(2,5,6,7)(9,10,14,15)(11,13,17,22)(12,19,21,16)(18,24,23,20)(25,27,31,29)(26,32,28,33)(30,35,38,37)(34,39,36,40);(2,6)(5,7)(10,15)(13,22)(16,19)(20,24)(25,27,31,29)(26,33,28,32)(30,35,38,37)(34,40,36,39);(1,3)(2,5)(4,8)(6,7)(9,11,14,17)(10,13,15,22)(12,18,21,23)(16,20,19,24)(25,28,31,26)(27,32,29,33)(30,36,38,34)(35,39,37,40);(1,3)(2,5)(4,8)(6,7)(9,12,14,21)(10,16,15,19)(11,18,17,23)(13,20,22,24)(25,29,31,27)(26,32,28,33)(30,37,38,35)(34,39,36,40);(1,2)(3,5)(4,6)(7,8)(9,13)(10,17)(11,15)(12,20)(14,22)(16,23)(18,19)(21,24)(25,30,31,38)(26,34,28,36)(27,35,29,37)(32,39,33,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=ghi,e^2=ghi,f^2=gh,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, 8137902);