#8183830

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	#38950	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4405 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=h,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, 8091026);