#10181054

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	#39869	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4411 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=ghi,d^2=hi,e^2=gi,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, 10088250);