#8374179

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

4277 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,d^2=h,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,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, 8281375);