#8077283

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

4720 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,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, 7984479);