#7941678

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4133 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)(11,19)(12,22)(13,20)(14,18)(15,21)(16,23)(17,24)(26,32)(27,29)(33,39)(35,36);(1,2)(3,5)(4,6)(7,8)(9,11,14,21)(10,15,18,19)(12,17,23,20)(13,16,24,22)(25,26)(27,30)(28,33)(29,34)(31,32)(35,37)(36,40)(38,39);(9,12,14,23)(10,16,18,22)(11,20,21,17)(13,19,24,15)(25,27,31,29)(26,30,32,34)(28,36,38,35)(33,40,39,37);(9,12)(10,16)(11,17)(13,19)(14,23)(15,24)(18,22)(20,21)(25,28,31,38)(26,33,32,39)(27,35,29,36)(30,37,34,40);(1,3,4,7)(2,5,6,8)(9,13)(10,17)(11,16)(12,19)(14,24)(15,23)(18,20)(21,22)(25,29,31,27)(26,34,32,30)(28,36,38,35)(33,40,39,37);(9,10)(11,15)(12,16)(13,20)(14,18)(17,24)(19,21)(22,23)(25,30,31,34)(26,27,32,29)(28,37,38,40)(33,35,39,36)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=hi,e^2=gi,f^2=hi,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, 7848874);