#10040591

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4095 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9947787);