#7905470

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4525 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gi,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7812666);