#7763285

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4226 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)(3,7,8,5)(9,10,14,15)(11,12,17,20)(13,16,21,22)(18,19,23,24)(25,26,31,34)(27,28,32,29)(30,33,39,40)(35,36,38,37);(1,3,4,8)(2,5,6,7)(9,11,14,17)(10,12,15,20)(13,18,21,23)(16,19,22,24)(27,32)(28,29)(35,38)(36,37);(10,15)(11,17)(13,21)(19,24)(25,27)(26,28)(29,34)(30,38)(31,32)(33,37)(35,39)(36,40);(9,11)(10,12)(13,18)(14,17)(15,20)(16,19)(21,23)(22,24)(25,28,31,29)(26,32,34,27)(30,36,39,37)(33,38,40,35);(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23)(25,29,31,28)(26,27,34,32)(30,37,39,36)(33,35,40,38);(9,13,14,21)(10,16,15,22)(11,18,17,23)(12,19,20,24)(25,30,31,39)(26,33,34,40)(27,35,32,38)(28,36,29,37)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=hi,e^2=hi,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,d^c=dh,f^c=fi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7670481);