#7812070

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	#44063	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4503 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=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, 7719266);