#7986347

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

4393 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=hi,d^2=h,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7893543);