#8077411

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

4448 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7984607);