#8445843

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

4311 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=gh,e^2=i,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, 8353039);