#8178726

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	#37669	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4465 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=h,e^2=h,f^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, 8085922);