#7718465

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	#50114	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4211 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7625661);