#7932956

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4201 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)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26,32,38)(27,35,29,33)(28,36,30,34)(31,37,40,39);(1,2)(3,5)(4,6)(7,8)(25,27,32,29)(26,33,38,35)(28,39,30,37)(31,36,40,34);(10,14)(15,17)(16,21)(22,23)(25,28,32,30)(26,34,38,36)(27,37,29,39)(31,33,40,35);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,29,32,27)(26,35,38,33)(28,37,30,39)(31,34,40,36);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,30,32,28)(26,36,38,34)(27,37,29,39)(31,33,40,35);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,31)(26,37)(27,36)(28,33)(29,34)(30,35)(32,40)(38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=i,e^2=g,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, 7840152);