#8120007

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

4274 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=g,e^2=gi,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, 8027203);