#8077324

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

4687 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)(11,17)(12,18)(13,19)(14,16)(15,22)(20,21)(23,24)(25,26,31,30)(27,34,29,32)(28,33,37,38)(35,40,36,39);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,17)(18,24)(19,23)(25,27,31,29)(26,32,30,34)(28,36,37,35)(33,40,38,39);(10,16)(13,18)(15,17)(21,24)(27,29)(32,34)(35,36)(39,40);(9,11)(10,17)(12,23)(13,21)(14,22)(15,16)(18,24)(19,20)(25,28)(26,33)(27,35)(29,36)(30,38)(31,37)(32,39)(34,40);(1,2,5,4)(3,6,8,7)(9,12,14,19)(10,18,16,13)(11,20,22,23)(15,24,17,21)(25,29,31,27)(26,34,30,32)(28,35,37,36)(33,39,38,40);(1,4,5,2)(3,7,8,6)(9,13,14,18)(10,19,16,12)(11,21,22,24)(15,23,17,20)(25,30,31,26)(27,34,29,32)(28,33,37,38)(35,39,36,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,e^2=g,f^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, 7984520);