#7901265

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×#555	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4241 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)(9,10,14,17)(11,15,19,13)(12,20,22,16)(18,21,24,23)(25,26,32,36)(27,34,29,31)(28,35,30,33)(37,39,38,40);(1,3,4,8)(2,5,6,7)(25,27,32,29)(26,31,36,34)(28,38,30,37)(33,40,35,39);(9,11,14,19)(10,13,17,15)(12,18,22,24)(16,21,20,23)(25,28,32,30)(26,33,36,35)(27,37,29,38)(31,39,34,40);(9,11)(10,15)(12,18)(13,17)(14,19)(16,23)(20,21)(22,24)(25,29,32,27)(26,34,36,31)(28,37,30,38)(33,39,35,40);(9,12,14,22)(10,16,17,20)(11,18,19,24)(13,21,15,23)(25,30,32,28)(26,35,36,33)(27,37,29,38)(31,39,34,40);(9,13,14,15)(10,11,17,19)(12,21,22,23)(16,18,20,24)(25,31,32,34)(26,27,36,29)(28,39,30,40)(33,37,35,38)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=i,e^2=h,f^2=h,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, 7808461);