#8977213

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	#35918	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4428 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)(3,8,4,6)(9,10,14,16)(11,17,18,12)(13,15,21,22)(19,23,24,20)(25,26)(27,34)(28,36)(29,33)(30,31)(32,37)(35,39)(38,40);(3,4)(6,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,22)(23,24)(25,27,31,33)(26,29,30,34)(28,35,37,40)(32,39,36,38);(1,3)(2,6)(4,5)(7,8)(11,18)(12,17)(13,21)(15,22)(26,30)(28,37)(29,34)(35,40);(3,4)(6,8)(9,11,14,18)(10,12,16,17)(13,19,21,24)(15,20,22,23)(25,28,31,37)(26,32,30,36)(27,35,33,40)(29,39,34,38);(1,4,5,3)(2,6,7,8)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,29,31,34)(26,33,30,27)(28,38,37,39)(32,35,36,40);(1,2)(3,8)(4,6)(5,7)(9,13,14,21)(10,15,16,22)(11,19,18,24)(12,20,17,23)(25,30,31,26)(27,34,33,29)(28,36,37,32)(35,38,40,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=g,e^2=h,f^2=g,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, 8884409);