#8658054

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	#43269	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4339 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8565250);