#8991831

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	#51458	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4189 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=hi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8899027);