#8174338

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	#35293	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4531 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)(6,8)(9,10,15,12)(11,16,18,14)(13,20,22,17)(19,21,24,23)(25,26,31,32)(27,28,29,33)(30,38,39,34)(35,40,37,36);(3,7)(6,8)(9,11,15,18)(10,16,12,14)(13,19,22,24)(17,23,20,21)(26,32)(28,33)(34,38)(36,40);(1,2,5,4)(3,6,7,8)(10,12)(11,18)(17,20)(19,24)(25,27,31,29)(26,33,32,28)(30,35,39,37)(34,40,38,36);(1,3,5,7)(2,6,4,8)(9,12,15,10)(11,16,18,14)(13,17,22,20)(19,21,24,23)(25,28,31,33)(26,27,32,29)(30,36,39,40)(34,35,38,37);(1,4,5,2)(3,8,7,6)(9,13,15,22)(10,17,12,20)(11,19,18,24)(14,21,16,23)(25,29,31,27)(26,28,32,33)(30,37,39,35)(34,36,38,40);(9,14,15,16)(10,18,12,11)(13,21,22,23)(17,24,20,19)(25,30,31,39)(26,34,32,38)(27,35,29,37)(28,36,33,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=hi,d^2=h,e^2=h,f^2=gi,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^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, 8081534);