#8813989

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	#47949	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4248 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gi,d^2=gh,e^2=gh,f^2=hi,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, 8721185);