#8152205

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	#34909	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4287 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=g,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,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, 8059401);