#8914226

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	#57287	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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8821422);