#8127529

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	#49560	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4406 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)(3,8)(9,10,15,12)(11,16,13,19)(14,20,22,17)(18,24,21,23)(29,37)(31,38)(34,40)(36,39);(1,2,6,5)(3,7,8,4)(9,11,15,13)(10,16,12,19)(14,18,22,21)(17,23,20,24)(25,26,30,32)(27,33,35,28)(29,31,37,38)(34,39,40,36);(3,8)(4,7)(10,12)(11,13)(17,20)(18,21)(25,27,30,35)(26,28,32,33)(29,34,37,40)(31,36,38,39);(1,3,6,8)(2,7,5,4)(9,12,15,10)(11,16,13,19)(14,20,22,17)(18,23,21,24)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,39)(38,40);(1,4)(2,3)(5,8)(6,7)(9,13,15,11)(10,16,12,19)(14,21,22,18)(17,23,20,24)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,39)(38,40);(1,5,6,2)(3,4,8,7)(9,14)(10,17)(11,18)(12,20)(13,21)(15,22)(16,23)(19,24)(25,29)(26,31)(27,34)(28,36)(30,37)(32,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=hi,d^2=gh,e^2=h,f^2=g,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^c=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, 8034725);