#7905428

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

4437 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=ghi,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7812624);