#10166641

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

4439 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=i,e^2=i,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, 10073837);