#10107007

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

4447 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)(11,12)(13,21)(14,15)(16,18)(17,23)(19,22)(20,24)(25,26,31,32)(27,28,29,33)(30,38,39,34)(35,40,37,36);(1,3,5,7)(2,4,8,6)(10,15)(12,16)(17,21)(20,22)(26,32)(27,29)(34,38)(35,37);(1,3)(2,6)(4,8)(5,7)(10,15)(11,18)(17,21)(19,24)(25,27,31,29)(26,33,32,28)(30,35,39,37)(34,40,38,36);(1,3,5,7)(2,4,8,6)(9,11,14,18)(10,16,15,12)(13,19,23,24)(17,22,21,20)(25,28,31,33)(26,27,32,29)(30,36,39,40)(34,35,38,37);(1,3)(2,4)(5,7)(6,8)(9,12,14,16)(10,11,15,18)(13,22,23,20)(17,24,21,19)(25,29,31,27)(26,33,32,28)(30,37,39,35)(34,40,38,36);(1,4)(2,7)(3,8)(5,6)(9,13,14,23)(10,17,15,21)(11,19,18,24)(12,20,16,22)(25,30,31,39)(26,34,32,38)(27,35,29,37)(28,36,33,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=ghi,d^2=h,e^2=g,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, 10014203);