#8075541

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

4458 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gi,d^2=h,f^2=h,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, 7982737);