#8127458

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	#2656	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4224 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,d^2=g,e^2=h,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, 8034654);