#8097080

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×#548	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4277 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)(3,7)(4,5)(6,8)(10,15)(11,17)(18,23)(19,24)(25,26,31,32)(27,33,28,29)(30,34,39,40)(35,38,36,37);(2,4)(6,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,21)(20,24)(22,23)(26,32)(27,28)(34,40)(35,36);(1,3,5,8)(2,6,4,7)(25,27,31,28)(26,29,32,33)(30,35,39,36)(34,37,40,38);(1,4,5,2)(3,7,8,6)(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,21,23,20)(25,28,31,27)(26,33,32,29)(30,36,39,35)(34,38,40,37);(9,12,14,16)(10,17,15,11)(13,21,22,20)(18,19,23,24)(25,29,31,33)(26,27,32,28)(30,38,39,37)(34,36,40,35);(9,13,14,22)(10,18,15,23)(11,19,17,24)(12,20,16,21)(25,30,31,39)(26,34,32,40)(27,35,28,36)(29,37,33,38)

Polycyclic presentation:

2^9|a^2=i,c^2=h,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, 8004276);