#8889388

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4145 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8796584);