#8073094

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

4442 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,e^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,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, 7980290);