#7853359

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4267 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)(11,19)(12,23)(13,17)(14,18)(15,22)(16,21)(20,24)(25,26,32,36)(27,33,29,31)(28,35,30,34)(37,39,38,40);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,24)(17,23)(18,19)(25,27,32,29)(26,33,36,31)(28,38,30,37)(34,39,35,40);(9,10,14,18)(11,15,22,19)(12,16,21,23)(13,17,20,24)(25,28,32,30)(26,34,36,35)(27,37,29,38)(31,39,33,40);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,24)(21,23)(25,29,32,27)(26,31,36,33)(28,37,30,38)(34,40,35,39);(9,12,14,21)(10,16,18,23)(11,20,22,13)(15,24,19,17)(25,30,32,28)(26,35,36,34)(27,37,29,38)(31,39,33,40);(1,4,5,3)(2,8,6,7)(9,13,14,20)(10,17,18,24)(11,21,22,12)(15,23,19,16)(25,31)(26,27)(28,39)(29,36)(30,40)(32,33)(34,37)(35,38)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=gh,e^2=h,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, 7760555);