#8195172

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	#59304	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4362 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=gh,e^2=h,f^2=gi,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, 8102368);