#10144540

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4200 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gh,e^2=gi,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, 10051736);