#8799447

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	#37379	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4303 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)(11,14)(12,20)(13,16)(15,21)(17,19)(18,23)(22,24)(26,31)(27,29)(33,36)(35,37);(1,2)(3,8)(4,5)(6,7)(9,10)(11,17)(12,20)(13,16)(14,19)(15,21)(18,22)(23,24)(25,26,30,31)(27,32,29,34)(28,36,38,33)(35,40,37,39);(2,7)(3,4)(9,10,13,16)(11,17,19,14)(12,15,21,20)(18,23,24,22)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,3,6,4)(2,8,7,5)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,24)(25,27,30,29)(26,32,31,34)(28,35,38,37)(33,39,36,40);(1,4,6,3)(2,5,7,8)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,28)(26,33)(27,35)(29,37)(30,38)(31,36)(32,39)(34,40);(1,5)(2,3)(4,7)(6,8)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,29,30,27)(26,34,31,32)(28,37,38,35)(33,40,36,39)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8706643);