#7739260

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	#37276	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4481 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7646456);