#8807292

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4164 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,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, 8714488);