#8183757

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	#38949	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4602 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)(11,15)(12,19)(13,14)(16,20)(17,23)(18,21)(22,24)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,39)(37,38);(2,3)(6,7)(9,10)(11,15)(12,16)(13,14)(17,22)(18,21)(19,20)(23,24)(25,27,31,28)(26,32,29,34)(30,35,38,36)(33,39,37,40);(1,3,5,2)(4,7,8,6)(10,14)(15,21)(16,19)(22,23)(25,28,31,27)(26,32,29,34)(30,35,38,36)(33,40,37,39);(1,4,5,8)(2,6,3,7)(9,11,13,18)(10,15,14,21)(12,17,20,24)(16,22,19,23)(25,29,31,26)(27,32,28,34)(30,37,38,33)(35,39,36,40);(9,12,13,20)(10,16,14,19)(11,17,18,24)(15,22,21,23);(9,10)(11,15)(12,16)(13,14)(17,22)(18,21)(19,20)(23,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=g,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8090953);