#8301625

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	#37738	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4209 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

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, 8208821);