#8202241

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	#36738	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4404 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,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, 8109437);