#8889325

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	#2635	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4120 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,d^2=g,e^2=gi,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, 8796521);