#8398611

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	#2433	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4438 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)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,26,31,33)(27,29,35,28)(30,32,38,40)(34,37,39,36);(1,2)(3,4)(5,6)(7,8)(9,11,15,12)(10,16,17,18)(13,22,24,20)(14,21,19,23)(25,27,31,35)(26,28,33,29)(30,34,38,39)(32,36,40,37);(2,6)(4,7)(10,17)(14,19)(16,18)(21,23)(25,28,31,29)(26,27,33,35)(30,37,38,36)(32,39,40,34);(2,6)(3,8)(9,12,15,11)(10,18,17,16)(13,20,24,22)(14,23,19,21)(25,29,31,28)(26,27,33,35)(30,37,38,36)(32,34,40,39);(1,3)(2,7)(4,6)(5,8)(9,13)(10,19)(11,20)(12,22)(14,17)(15,24)(16,21)(18,23)(25,27)(26,29)(28,33)(30,34)(31,35)(32,37)(36,40)(38,39);(1,4,5,7)(2,3,6,8)(9,14,15,19)(10,13,17,24)(11,21,12,23)(16,22,18,20)(25,30,31,38)(26,32,33,40)(27,34,35,39)(28,36,29,37)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^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, 8305807);