#7891208

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	#41140	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4375 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)(9,10,15,19)(11,18,13,16)(12,17,23,14)(20,21,22,24)(25,26)(27,32)(28,30)(29,39)(31,35)(33,38)(34,36)(37,40);(1,3,5,8)(2,6,4,7)(9,11,15,13)(10,16,19,18)(12,20,23,22)(14,21,17,24)(25,27,31,37)(26,32,35,40)(28,36,38,29)(30,34,33,39);(9,12,15,23)(10,14,19,17)(11,20,13,22)(16,21,18,24)(25,28,31,38)(26,33,35,30)(27,29,37,36)(32,34,40,39);(1,2,5,4)(3,6,8,7)(9,12)(10,17)(11,20)(13,22)(14,19)(15,23)(16,24)(18,21)(25,28)(26,30)(27,36)(29,37)(31,38)(32,34)(33,35)(39,40);(1,4,5,2)(3,7,8,6)(9,13,15,11)(10,18,19,16)(12,22,23,20)(14,24,17,21)(25,29,31,36)(26,34,35,39)(27,28,37,38)(30,32,33,40);(9,14,15,17)(10,12,19,23)(11,21,13,24)(16,20,18,22)(25,30)(26,28)(27,34)(29,40)(31,33)(32,36)(35,38)(37,39)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=ghi,e^2=gi,f^2=hi,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, 7798404);