#8445852

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4327 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,d^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8353048);