#10106972

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	#43530	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4366 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)(3,4)(5,8)(6,7)(9,10,15,17)(11,20,22,16)(12,23,19,14)(13,18,24,21)(26,32)(28,37)(33,34)(35,38);(1,3,5,7)(2,4,8,6)(9,11,15,22)(10,16,17,20)(12,21,19,18)(13,23,24,14)(25,26)(27,33)(28,30)(29,34)(31,32)(35,36)(37,40)(38,39);(1,3)(2,6)(4,8)(5,7)(10,17)(12,19)(16,20)(18,21)(25,27,31,29)(26,33,32,34)(28,35,37,38)(30,36,40,39);(1,3,5,7)(2,4,8,6)(9,12,15,19)(10,14,17,23)(11,21,22,18)(13,20,24,16)(25,28)(26,30)(27,35)(29,38)(31,37)(32,40)(33,36)(34,39);(1,3)(2,4)(5,7)(6,8)(9,13,15,24)(10,18,17,21)(11,14,22,23)(12,16,19,20)(25,29,31,27)(26,34,32,33)(28,35,37,38)(30,36,40,39);(1,4)(2,7)(3,8)(5,6)(9,14)(10,19)(11,13)(12,17)(15,23)(16,18)(20,21)(22,24)(25,30)(26,28)(27,36)(29,39)(31,40)(32,37)(33,35)(34,38)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,d^2=h,e^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, 10014168);