#8393270

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	#45454	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4483 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=i,e^2=g,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, 8300466);