#10060937

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	#2669	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4354 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)(25,26,31,28)(27,34,29,32)(30,33,38,36)(35,40,37,39);(1,3,5,8)(2,6,7,4)(10,14)(11,17)(16,19)(18,23)(25,27,31,29)(26,32,28,34)(30,35,38,37)(33,39,36,40);(2,7)(3,8)(10,14)(11,17)(16,19)(18,23)(25,26,31,28)(27,32,29,34)(30,33,38,36)(35,39,37,40);(1,3)(2,4)(5,8)(6,7)(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,28,31,26)(27,32,29,34)(30,36,38,33)(35,39,37,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,29,31,27)(26,32,28,34)(30,35,38,37)(33,40,36,39);(1,4,5,6)(2,8,7,3)(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=g,e^2=ghi,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, 9968133);