#8097277

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×#550	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4411 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=g,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004473);