#10173757

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
prime radical	#43837	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4343 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,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, 10080953);