#8108292

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	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4484 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015488);