#10195928

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

4301 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,d^2=h,e^2=gh,f^2=ghi,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, 10103124);