#7857919

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	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4271 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,d^2=h,e^2=gh,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, 7765115);