#8899058

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	#51084	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4327 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,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, 8806254);