#10460658

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	#49749	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4331 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10367854);