#10183178

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	#37245	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4259 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)(3,5)(4,6)(7,8)(9,10,15,14)(11,17,12,16)(13,18,21,22)(19,24,20,23)(25,26)(27,30)(28,33)(29,34)(31,32)(35,40)(36,38)(37,39);(1,3,7,6)(2,4,8,5)(9,11,15,12)(10,16,14,17)(13,19,21,20)(18,23,22,24)(25,27,31,35)(26,30,32,40)(28,36,37,29)(33,38,39,34);(1,4,7,5)(2,3,8,6)(26,32)(27,35)(28,37)(34,38);(2,8)(4,5)(9,12,15,11)(10,17,14,16)(13,19,21,20)(18,23,22,24)(25,28)(26,33)(27,36)(29,35)(30,38)(31,37)(32,39)(34,40);(1,5,7,4)(2,3,8,6)(9,13,15,21)(10,18,14,22)(11,19,12,20)(16,23,17,24)(25,29,31,36)(26,34,32,38)(27,37,35,28)(30,39,40,33);(1,6,7,3)(2,5,8,4)(9,14,15,10)(11,17,12,16)(13,18,21,22)(19,23,20,24)(25,30)(26,35)(27,32)(28,38)(29,33)(31,40)(34,37)(36,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=hi,d^2=ghi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10090374);