#10144531

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	#46397	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4310 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)(11,18)(12,16)(13,21)(14,15)(17,23)(19,20)(22,24)(25,26,32,34)(27,35,31,33)(28,30,29,39)(36,40,37,38);(1,2)(3,4)(5,7)(6,8)(10,15)(12,18)(17,21)(20,22)(25,27,32,31)(26,33,34,35)(28,36,29,37)(30,38,39,40);(1,3)(2,6)(4,7)(5,8)(10,15)(11,16)(17,21)(19,24)(25,28,32,29)(26,30,34,39)(27,36,31,37)(33,38,35,40);(1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,22)(23,24)(25,29,32,28)(26,30,34,39)(27,37,31,36)(33,38,35,40);(1,4,5,6)(2,3,7,8)(9,12)(10,16)(11,15)(13,22)(14,18)(17,19)(20,23)(21,24)(25,30)(26,29)(27,38)(28,34)(31,40)(32,39)(33,37)(35,36);(1,3)(2,4)(5,8)(6,7)(9,13,14,23)(10,17,15,21)(11,19,16,24)(12,20,18,22)(25,31,32,27)(26,33,34,35)(28,37,29,36)(30,38,39,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gh,e^2=i,f^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, 10051727);