#8172632

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	#59213	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4279 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,d^2=gh,e^2=gh,f^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, 8079828);