#8132499

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

4417 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=h,e^2=g,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^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, 8039695);