#7868916

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	#47760	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4457 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=ghi,e^2=h,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, 7776112);