#10316851

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	#53694	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4206 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,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, 10224047);