#8772490

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4167 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)(3,7)(9,10,16,22)(11,19,13,17)(12,21,15,18)(14,20,24,23)(27,29)(31,33)(35,37)(39,40);(1,2,6,5)(3,8,7,4)(9,11,16,13)(10,17,22,19)(12,23,15,20)(14,21,24,18)(25,26,30,34)(27,31,29,33)(28,36,38,32)(35,40,37,39);(1,3)(2,4)(5,8)(6,7)(9,12,16,15)(10,18,22,21)(11,23,13,20)(14,19,24,17)(25,27,30,29)(26,31,34,33)(28,35,38,37)(32,39,36,40);(1,2,6,5)(3,4,7,8)(9,13,16,11)(10,19,22,17)(12,23,15,20)(14,21,24,18);(1,4,6,8)(2,7,5,3)(9,14,16,24)(10,20,22,23)(11,18,13,21)(12,17,15,19)(25,28,30,38)(26,32,34,36)(27,35,29,37)(31,39,33,40);(1,5,6,2)(3,8,7,4)(9,15,16,12)(10,21,22,18)(11,23,13,20)(14,19,24,17)(25,29,30,27)(26,33,34,31)(28,37,38,35)(32,40,36,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=hi,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,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, 8679686);