#8943196

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	#53608	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4467 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,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, 8850392);