#7852770

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	#2541	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4396 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=h,e^2=g,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, 7759966);