#8649967

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	#51890	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4471 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=gi,d^2=i,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8557163);