#8936656

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	#37720	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4549 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi

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, 8843852);