#10166630

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

4607 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=i,e^2=gi,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, 10073826);