#7888967

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	#2661	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4239 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)(11,13)(16,18)(19,21)(23,24)(25,26,30,34)(27,33,38,31)(28,32,39,29)(35,40,37,36);(1,2)(3,7)(4,8)(5,6)(9,10,15,12)(11,18,13,16)(14,17,22,20)(19,24,21,23)(25,27)(26,31)(28,37)(29,36)(30,38)(32,40)(33,34)(35,39);(9,11,15,13)(10,16,12,18)(14,19,22,21)(17,23,20,24)(25,28)(26,32)(27,35)(29,34)(30,39)(31,36)(33,40)(37,38);(1,3,5,4)(2,7,6,8)(9,12,15,10)(11,16,13,18)(14,20,22,17)(19,23,21,24)(25,29,30,32)(26,28,34,39)(27,36,38,40)(31,37,33,35);(1,3,5,4)(2,7,6,8)(9,13,15,11)(10,16,12,18)(14,19,22,21)(17,24,20,23)(25,28)(26,32)(27,37)(29,34)(30,39)(31,40)(33,36)(35,38);(1,4,5,3)(2,8,6,7)(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,27)(26,33)(28,35)(29,36)(30,38)(31,34)(32,40)(37,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=gh,e^2=gi,f^2=ghi,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, 7796163);