#7714457

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	#39871	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4169 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)(9,10,15,18)(11,19,20,16)(12,14,23,24)(13,17,21,22)(25,26)(27,31)(28,36)(29,34)(30,32)(33,38)(35,40)(37,39);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,13)(14,22)(15,20)(17,24)(18,19)(21,23)(25,27,30,29)(26,31,32,34)(28,35,38,37)(33,39,36,40);(11,20)(13,21)(16,19)(17,22)(26,32)(31,34)(33,36)(39,40);(1,2,5,4)(3,6,8,7)(9,12)(10,14)(11,21)(13,20)(15,23)(16,17)(18,24)(19,22)(25,26)(27,31)(28,33)(29,34)(30,32)(35,39)(36,38)(37,40);(1,4,5,2)(3,7,8,6)(9,13,15,21)(10,17,18,22)(11,12,20,23)(14,16,24,19)(25,28)(26,33)(27,35)(29,37)(30,38)(31,39)(32,36)(34,40);(9,14)(10,12)(11,22)(13,16)(15,24)(17,20)(18,23)(19,21)(25,29,30,27)(26,34,32,31)(28,37,38,35)(33,40,36,39)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7621653);