#8108724

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	C₂×#537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4376 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)(3,8)(4,7)(5,6)(9,10,14,16)(11,17,18,12)(13,21,22,15)(19,20,24,23)(28,35)(32,37)(34,39)(38,40);(2,6)(7,8)(11,18)(12,17)(13,22)(15,21)(26,30)(27,33)(32,37)(34,39);(9,11)(10,12)(13,19)(14,18)(15,20)(16,17)(21,23)(22,24)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,3,5,4)(2,7,6,8)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,24)(22,23);(1,2)(3,7)(4,8)(5,6)(9,13,14,22)(10,15,16,21)(11,19,18,24)(12,20,17,23)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(1,4,5,3)(2,8,6,7)(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,22)(23,24)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015920);