#8758023

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

4513 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,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, 8665219);