#8120035

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	#40308	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4206 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:

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

Polycyclic presentation:

2^9|b^2=hi,d^2=h,e^2=i,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^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, 8027231);