#8746943

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	#44987	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4335 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8654139);