#7893522

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4197 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=i,e^2=g,f^2=h,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, 7800718);