#8399431

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	#34962	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4379 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8306627);