#7853313

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4288 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7760509);