#8165366

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	#59854	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4320 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8072562);