#8069753

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

4585 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=g,e^2=g,f^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, 7976949);