#7658673

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

4083 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)(9,10,16,15)(11,19,20,14)(12,21,23,17)(13,18,22,24)(25,26,30,31)(27,29,34,32)(28,36,38,33)(35,39,40,37);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,18)(13,21)(15,19)(16,20)(17,22)(23,24)(26,31)(27,34)(33,36)(35,40);(9,12,16,23)(10,17,15,21)(11,18,20,24)(13,14,22,19)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,39,36,37);(1,2,5,4)(3,6,8,7)(9,13,16,22)(10,18,15,24)(11,17,20,21)(12,14,23,19)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(1,4,5,2)(3,7,8,6)(9,14)(10,11)(12,22)(13,23)(15,20)(16,19)(17,18)(21,24)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,40,36,35);(9,15,16,10)(11,19,20,14)(12,21,23,17)(13,24,22,18)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=g,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7565869);