#7719412

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	#2668	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4343 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=hi,e^2=hi,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, 7626608);