#8665024

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	#55424	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=ghi,e^2=h,f^2=g,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, 8572220);