#8152172

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

4342 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)(10,12)(11,17)(13,21)(22,24)(25,26,30,31)(27,29,34,32)(28,36,38,33)(35,39,40,37);(3,7)(6,8)(9,10,14,12)(11,18,17,15)(13,16,21,20)(19,24,23,22)(26,31)(27,34)(33,36)(35,40);(1,2)(3,6)(4,5)(7,8)(9,10,14,12)(11,15,17,18)(13,16,21,20)(19,22,23,24)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,39,36,37);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,15,17,18)(13,20,21,16)(19,22,23,24)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,40,36,35)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=g,e^2=g,f^2=i,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, 8059368);