#7954544

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4120 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)(11,15)(12,20)(13,17)(14,18)(16,22)(19,24)(21,23)(26,28)(27,29)(33,36)(35,37);(1,2,5,6)(3,7,4,8)(9,11,14,13)(10,15,18,17)(12,19,22,21)(16,23,20,24)(25,26,31,28)(27,34,29,32)(30,33,38,36)(35,40,37,39);(1,3,5,4)(2,7,6,8)(9,10,14,18)(11,15,13,17)(12,16,22,20)(19,23,21,24)(25,27,31,29)(26,32,28,34)(30,35,38,37)(33,39,36,40);(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,22)(21,24)(25,28,31,26)(27,32,29,34)(30,36,38,33)(35,39,37,40);(1,3,5,4)(2,7,6,8)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20)(25,29,31,27)(26,32,28,34)(30,35,38,37)(33,40,36,39);(1,4,5,3)(2,8,6,7)(9,13,14,11)(10,17,18,15)(12,21,22,19)(16,24,20,23)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^9|b^2=ghi,c^2=ghi,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7861740);