#10349109

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

4235 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)(3,4)(5,6)(7,8)(9,10,14,12)(11,15,19,17)(13,20,21,16)(18,23,24,22)(25,26,30,33)(27,28,31,36)(29,32,38,40)(34,35,37,39);(2,6)(3,8)(9,10,14,12)(11,17,19,15)(13,20,21,16)(18,22,24,23)(25,27,30,31)(26,28,33,36)(29,37,38,34)(32,39,40,35);(2,6)(4,7)(10,12)(15,17)(16,20)(22,23)(25,28,30,36)(26,31,33,27)(29,35,38,39)(32,37,40,34);(1,3)(2,7)(4,6)(5,8)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27)(26,28)(29,34)(30,31)(32,35)(33,36)(37,38)(39,40);(1,4,5,7)(2,3,6,8)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,28)(26,31)(27,33)(29,35)(30,36)(32,37)(34,40)(38,39);(1,3)(2,7)(4,6)(5,8)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,29)(26,32)(27,34)(28,35)(30,38)(31,37)(33,40)(36,39)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ghi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10256305);