#8202304

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	#36758	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4398 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)(3,7)(4,6)(5,8)(9,10,14,18)(11,15,19,24)(12,16,22,21)(13,17,23,20)(25,26,31,34)(27,33,28,32)(29,39,40,30)(35,38,37,36);(1,2,4,6)(3,5,8,7)(11,19)(12,22)(15,24)(16,21)(25,27,31,28)(26,32,34,33)(29,37,40,35)(30,36,39,38);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,20)(13,16)(14,19)(17,22)(18,24)(21,23)(25,28,31,27)(26,33,34,32)(29,37,40,35)(30,36,39,38);(1,3,4,8)(2,5,6,7)(25,29)(26,30)(27,35)(28,37)(31,40)(32,38)(33,36)(34,39);(9,12)(10,16)(11,20)(13,15)(14,22)(17,19)(18,21)(23,24)(25,29)(26,30)(27,35)(28,37)(31,40)(32,38)(33,36)(34,39);(9,13)(10,17)(11,21)(12,15)(14,23)(16,19)(18,20)(22,24)(25,30)(26,29)(27,36)(28,38)(31,39)(32,37)(33,35)(34,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8109500);