#8899564

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	#51567	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4307 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8806760);