#7891210

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4465 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,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, 7798406);