#8963364

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

4658 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)(3,5)(4,7)(6,8)(9,10,14,16)(11,15,18,12)(13,21,22,17)(19,20,24,23)(25,26,30,28)(27,31,33,34)(29,32,37,36)(35,38,39,40);(1,2,6,8)(3,5,7,4)(9,11,14,18)(10,15,16,12)(13,19,22,24)(17,23,21,20)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,3,6,7)(2,4,8,5)(10,16)(11,18)(13,22)(20,23)(27,33)(29,37)(31,34)(32,36);(1,4,6,5)(2,3,8,7)(9,12)(10,11)(13,20)(14,15)(16,18)(17,19)(21,24)(22,23)(25,28,30,26)(27,34,33,31)(29,32,37,36)(35,38,39,40);(1,2)(3,4)(5,7)(6,8)(9,11)(10,15)(12,16)(13,19)(14,18)(17,23)(20,21)(22,24)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40);(1,5,6,4)(2,7,8,3)(9,13,14,22)(10,17,16,21)(11,19,18,24)(12,20,15,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=ghi,e^2=hi,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, 8870560);