#7812102

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4426 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=g,e^2=g,f^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, 7719298);