#8528929

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

4531 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8436125);