#8172716

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	#38032	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4300 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gi,d^2=g,e^2=g,f^2=i,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, 8079912);