#10106442

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

4519 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)(5,7)(6,8)(9,10,14,15)(11,16,19,22)(12,17,23,20)(13,21,24,18)(25,26)(27,30)(28,35)(29,32)(31,36)(33,39)(34,38)(37,40);(1,3)(2,6)(4,7)(5,8)(9,10,14,15)(11,16,19,22)(12,20,23,17)(13,18,24,21)(25,27,29,34)(26,30,32,38)(28,33,36,40)(31,37,35,39);(10,15)(11,19)(17,20)(18,21)(25,26,29,32)(27,30,34,38)(28,31,36,35)(33,37,40,39);(2,7)(4,6)(9,11)(10,16)(12,21)(13,20)(14,19)(15,22)(17,24)(18,23)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(1,4,5,6)(2,3,7,8)(9,12)(10,17)(11,18)(13,16)(14,23)(15,20)(19,21)(22,24)(25,27)(26,30)(28,33)(29,34)(31,37)(32,38)(35,39)(36,40);(1,3)(2,4)(5,8)(6,7)(9,13)(10,18)(11,20)(12,22)(14,24)(15,21)(16,23)(17,19)(25,28)(26,31)(27,33)(29,36)(30,37)(32,35)(34,40)(38,39)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,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, 10013638);