#8818169

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4262 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=gi,e^2=gh,f^2=gi,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, 8725365);