#8936723

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

4373 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:

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

Polycyclic presentation:

2^9|c^2=h,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=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, 8843919);