#8798619

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	#43275	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4200 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=gi,e^2=ghi,f^2=g,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, 8705815);