#10106974

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=ghi,d^2=h,e^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, 10014170);