#10097057

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	#52005	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4512 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)(11,16)(12,13)(14,21)(15,17)(18,20)(19,23)(22,24)(25,26)(27,35)(28,38)(29,32)(30,34)(31,33)(36,39)(37,40);(1,2,5,8)(3,4,7,6)(9,11)(10,16)(12,20)(13,18)(14,22)(15,19)(17,23)(21,24)(25,27,32,31)(26,33,29,35)(28,37,30,36)(34,40,38,39);(1,3)(2,6)(4,8)(5,7)(10,17)(11,19)(12,22)(18,21)(25,28,32,30)(26,34,29,38)(27,36,31,37)(33,39,35,40);(1,4)(2,7)(3,8)(5,6)(9,12,15,22)(10,13,17,24)(11,20,19,14)(16,18,23,21)(25,29,32,26)(27,35,31,33)(28,34,30,38)(36,39,37,40);(1,3)(2,4)(5,7)(6,8)(9,13)(10,12)(11,21)(14,16)(15,24)(17,22)(18,19)(20,23)(25,30,32,28)(26,34,29,38)(27,37,31,36)(33,39,35,40);(1,4)(2,7)(3,8)(5,6)(9,14,15,20)(10,18,17,21)(11,22,19,12)(13,23,24,16)(25,31,32,27)(26,35,29,33)(28,37,30,36)(34,40,38,39)

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,d^2=h,e^2=g,f^2=h,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, 10004253);