#8117974

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	#38388	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4439 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8025170);