#8761828

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	#46372	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4535 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,e^2=g,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, 8669024);