#7900928

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
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4169 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,d^2=h,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, 7808124);