#9947475

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	#37825	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4386 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,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, 9854671);