#9947466

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	#43012	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4440 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)(9,10,15,17)(11,14,12,16)(13,18,22,24)(19,21,20,23)(25,26,32,34)(27,28,29,30)(31,39,40,33)(35,38,37,36);(1,2)(3,4)(5,7)(6,8)(9,11,15,12)(10,16,17,14)(13,20,22,19)(18,21,24,23)(25,27)(26,28)(29,32)(30,34)(31,35)(33,36)(37,40)(38,39);(1,2)(3,4)(5,7)(6,8)(10,17)(11,12)(18,24)(19,20)(25,28,32,30)(26,27,34,29)(31,36,40,38)(33,35,39,37);(1,3)(2,6)(4,7)(5,8)(9,12,15,11)(10,14,17,16)(13,20,22,19)(18,21,24,23)(25,29,32,27)(26,28,34,30)(31,37,40,35)(33,36,39,38);(1,4,5,6)(2,3,7,8)(9,13,15,22)(10,18,17,24)(11,19,12,20)(14,21,16,23)(25,30,32,28)(26,27,34,29)(31,36,40,38)(33,37,39,35);(1,3)(2,4)(5,8)(6,7)(9,14)(10,11)(12,17)(13,21)(15,16)(18,19)(20,24)(22,23)(25,31)(26,33)(27,35)(28,36)(29,37)(30,38)(32,40)(34,39)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gh,d^2=i,e^2=h,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, 9854662);