#9713688

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	#2540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4265 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)(4,5)(6,8)(10,13)(15,17)(16,21)(22,23)(25,26,31,35)(27,34,30,32)(28,29,33,40)(36,37,38,39);(3,8)(6,7)(12,20)(16,21)(18,24)(22,23)(25,27,31,30)(26,32,35,34)(28,36,33,38)(29,39,40,37);(1,3)(2,6)(4,7)(5,8)(9,10,14,13)(11,17,19,15)(12,16,20,21)(18,23,24,22)(25,27,31,30)(26,32,35,34)(28,38,33,36)(29,37,40,39);(1,2,5,4)(3,7,8,6)(9,11,14,19)(10,15,13,17)(12,18,20,24)(16,22,21,23)(25,28,31,33)(26,29,35,40)(27,36,30,38)(32,39,34,37);(1,3)(2,7)(4,6)(5,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,29)(26,33)(27,37)(28,35)(30,39)(31,40)(32,36)(34,38);(1,4,5,2)(3,7,8,6)(9,13,14,10)(11,17,19,15)(12,21,20,16)(18,23,24,22)(25,30,31,27)(26,34,35,32)(28,38,33,36)(29,37,40,39)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

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, 9620884);