#8762120

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	#45438	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4940 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8669316);