#8914113

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	#45917	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4355 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=gh,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8821309);