#8947266

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	#35697	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4277 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8854462);