#8889205

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	#35095	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4209 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8796401);