#8936262

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	#35696	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4364 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8843458);