#10106990

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

4456 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)(3,4)(5,8)(6,7)(9,10,14,15)(11,17,18,12)(13,21,23,16)(19,20,24,22)(25,26)(27,36)(28,33)(29,40)(30,34)(31,35)(32,39)(37,38);(1,3,5,7)(2,4,8,6)(10,15)(12,17)(16,21)(20,22)(25,27,31,39)(26,32,35,36)(28,37,40,34)(29,38,33,30);(1,3)(2,6)(4,8)(5,7)(10,15)(11,18)(16,21)(19,24)(25,28,31,40)(26,33,35,29)(27,37,39,34)(30,36,38,32);(1,3,5,7)(2,4,8,6)(9,11,14,18)(10,12,15,17)(13,19,23,24)(16,20,21,22)(25,29)(26,28)(27,38)(30,39)(31,33)(32,37)(34,36)(35,40);(1,3)(2,4)(5,7)(6,8)(9,12)(10,11)(13,22)(14,17)(15,18)(16,24)(19,21)(20,23)(25,26)(27,32)(28,33)(29,40)(30,37)(31,35)(34,38)(36,39);(1,4)(2,7)(3,8)(5,6)(9,13,14,23)(10,16,15,21)(11,19,18,24)(12,20,17,22)(25,30,31,38)(26,34,35,37)(27,29,39,33)(28,36,40,32)

Polycyclic presentation:

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

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, 10014186);