#8195829

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4296 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)(11,14)(12,16)(13,21)(15,18)(17,23)(19,24)(20,22)(26,28)(27,33)(32,36)(35,37);(1,2,5,4)(3,7,8,6)(9,10)(11,14)(12,16)(13,17)(15,18)(19,22)(20,24)(21,23)(25,26,30,28)(27,31,33,34)(29,32,38,36)(35,39,37,40);(2,4)(6,7)(9,11,15,12)(10,16,18,14)(13,19,23,20)(17,24,21,22)(25,27)(26,31)(28,34)(29,37)(30,33)(32,40)(35,38)(36,39);(1,3)(2,6)(4,7)(5,8)(9,12,15,11)(10,16,18,14)(13,20,23,19)(17,24,21,22)(25,26,30,28)(27,31,33,34)(29,32,38,36)(35,39,37,40);(9,13,15,23)(10,17,18,21)(11,19,12,20)(14,22,16,24)(25,28,30,26)(27,34,33,31)(29,36,38,32)(35,40,37,39);(1,4,5,2)(3,7,8,6)(9,14,15,16)(10,11,18,12)(13,22,23,24)(17,19,21,20)(25,29,30,38)(26,32,28,36)(27,35,33,37)(31,39,34,40)

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,d^2=h,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, 8103025);