#8301761

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	#47851	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4276 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8208957);